higher outcome 2 higher unit 3 further differentiation trig functions harder type questions further...
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Higher Outcome 2
Higher Unit 3Higher Unit 3
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Further Differentiation Trig Functions
Harder Type Questions
Further Integration
Integration Exam Type Questions
Integrating Trig Functions
Differentiation The Chain Rule
Harder Type Questions
Differentiation Exam Type Questions
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Higher Outcome 2
Trig Function Differentiation
The Derivatives of sin x & cos x
(sin ) cos (cos ) sin d d
x x x xdx dx
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Higher Outcome 2Example
2( ) 2cos sin '( )
3 . Find f x x x f x
2'( ) (2cos ) sin
3
d d
f x x xdx dx
[ ( ) ( )]d df dgf x g x
dx dx dx
22 (cos ) sin
3
d dx x
dx dx. ( )
d dgc g x c
dx dx
2'( ) 2sin cos
3 f x x x
Trig Function Differentiation
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Higher Outcome 2Example
3
3
4 cos Differentiate with respect to
x xx
x
33
3
4 cos4 cos
x x
x xx
3 3(4 cos ) 4 (cos ) d d d
x x x xdx dx dx
4
4
12 ( sin )
12 sin
x x
x x
4
4
sin 12 x x
x
Trig Function Differentiation
Simplify expression -
where possible
Restore the original form of expression
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Higher Outcome 2
The Chain Rule for Differentiating
To differentiate composite functions (such as functions with brackets in them) use:
dy dy du
dx du dx
Example 2 8( 3) Differentiate y x
2( 3) Let u x 8 then y u
78 dy
udu
2 du
xdx
dy dy du
dx du dx78 2 u x 2 78( 3) .2 x x 2 716 ( 3)
dyx x
dx
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Higher Outcome 2
The Chain Rule for DifferentiatingTrig Functions
Trig functions can be differentiated using the same chain rule method.
dy dy du
dx du dx
Worked Example:
sin 4 Differentiate y x
4 Let u x sin Then y u
4 du
dxcos
dyu
du
dy dy du
dx du dxcos 4 u 4cos4
dyx
dx
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Higher Outcome 2
The Chain Rule for Differentiating
Example
42
5'( ) ( )
3
Find when f x f x
x
2( 3) Let u x4( ) 5 Then f x u
520 df
udu
2 du
xdx
520 2 u x 2 520( 3) 2 x x
2 5
40
( 3)
df x
dx x
df df du
dx du dx
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Higher Outcome 2Example
1
2 1
Find when
dyy
dx x
(2 1) Let u x1
2
Then y u
2du
dx
3
21
2
dyu
du
dy dy du
dx du dx
3
21
22
u
3
2
u
3
2
1
(2 1)
dy
dxx
The Chain Rule for Differentiating
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Higher Outcome 2Example 3 2 5( ) (3 2 1) '(0) If , find the value of f x x x f
3 2(3 2 1) Let u x x 5( ) Then f x u
2(9 4 ) du
x xdx
45 df
udu
df df du
dx du dx4 25 (9 4 ) u x x 3 2 4 25(3 2 1) (9 4 ) x x x x
'(0) 0 f
The Chain Rule for Differentiating
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Higher Outcome 2
The Chain Rule for DifferentiatingTrig Functions
Example
4cos Find when dy
y xdx
cos Let u x 4 Then y u
sin du
xdx
34 dy
udu
dy dy du
dx du dx34 ( sin ) u x
34sin cos dy
x xdx
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Higher Outcome 2Example
sin Find when dy
y xdx
sin Let u x1
2 Then y u
cos du
xdx
1
21
2
dyu
du
dy dy du
dx du dx
1
21
cos2
u x
cos
2 sin
dy x
dx x
The Chain Rule for DifferentiatingTrig Functions
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Higher Outcome 2Example1
32 1
Find the equation of the tangent to the curve where y xx
1
2 1
y
xRe-arrange: 1
2 1 y x
The slope of the tangent is given by the derivative of the equation.
Use the chain rule:
21 2 1 2
dy
xdx 2
2
2 1
x
Where x = 3:
1 2
5 25
and
dyy
dx
1 23,
5 25
We need the equation of the line through with slope
The Chain Rule for Differentiating Functions
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Higher Outcome 2
1 2( 3)
5 25
y x
2 1 1 5( 3) (2 6)
25 5 25 25
y x x
1(2 1)
25
y x Is the required equation
Remember
y - b = m(x – a)
The Chain Rule for Differentiating Functions
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Higher Outcome 2Example
In a small factory the cost, C, in pounds of assembling x components in a month is given by:
21000
( ) 40 , 0
C x x xx
Calculate the minimum cost of production in any month, and the corresponding number of
components that are required to be assembled.
0 Minimum occurs where , so differentiate dC
dx
225
( ) 40
C x xx
Re-arrange 2
251600
x
x
The Chain Rule for Differentiating Functions
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Higher Outcome 2
225
1600
dC d
xdx dx x
225
1600
dx
dx x
Using chain rule 2
25 251600 2 1
x
x x 2
25 253200 1
x
x x
2
25 250 0 1 0
where or dC
xdx x x
2 225 25 i.e. where or x x
2 25No real values satisfy x 0 We must have x
5 We must have items assembled per month x
The Chain Rule for Differentiating Functions
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Higher Outcome 2
2
25 253200 1 0 5
when dC
x xdx x x
5 5 Consider the sign of when and dC
x xdx
For x < 5 we have (+ve)(+ve)(-ve) = (-ve)
Therefore x = 5 is a minimum
Is x = 5 a minimum in the (complicated) graph?
Is this a minimum
?
For x > 5 we have (+ve)(+ve)(+ve) = (+ve)
The Chain Rule for Differentiating Functions
For x = 5 we have (+ve)(+ve)(0) = 0x = 5
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Higher Outcome 2
The cost of production:2
25( ) 1600 5
, C x x x
x
225
1600 55
21600 10
160,000 £ Expensive components?
Aeroplane parts maybe ?
The Chain Rule for Differentiating Functions
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Higher Outcome 2
Integrating Composite Functions
By “reversing” the Chain Rule we see that:
1( )( )
( 1)
n
n ax bax b dx c
a n
Worked Example
4(3 7) x dx5(3 7)
15
xc
4 1(3 7)
3(4 1)
x
c
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Higher Outcome 2
Integrating Composite Functions
1( )( )
( 1)
n
n ax bax b dx c
a n
Example : 4(4 2) Evaluate dt
t
4(4 2) t dt4 1(4 2)
4( 4 1)
tc
3(4 2)
12
tc
Re-arrangeCompare with standard form
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Higher Outcome 2Example
2( 3) Evaluate u du
Compare with standard form
2( 3) u du2 1( 3)
1( 2 1)
uc
1( 3)
1
uc
2 1( 3)
( 3)
u du cu
Integrating Composite Functions
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Higher Outcome 2Example 2 3
1(2 1) Evaluate x dx
3(2 1) x dx3 1(2 1)
( )2(3 1)
xc( ) F x
4(2 1)( )
8
xc
2 3
1(2 1) x dx (2) (1) F F 4 41
(5 3 )8
2 2 2 21(5 3 )(5 3 )
8
1(16)(34)
8
2 3
1(2 1) 68 x dx
Integrating Composite Functions
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Higher Outcome 2Example
32
4
1(3 4)
Evaluate x dx
32(3 4) x dx ( ) F x
32
1
32
(3 4)( )
3( 1)
x
c522(3 4)
( )15
xc
32
4
1(3 4)
x dx (4) ( 1) F F
5 52 2
2(16 (1) )
15 52
(4 1)15
2(1024 1)
15
2046
15
32
4
1
2046(3 4)
5x dx
Integrating Composite Functions
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Higher Outcome 2
Integrating Trig Functions
(sin ) cos
(cos ) sin
dx x
dxd
x xdx
cos sin
sin cos
x dx x c
x dx x c
Integration is opposite of
differentiation
Worked Example
123sin cosx x dx 3cos x 1
2 sin x c
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Higher Outcome 2
From “reversing” the chain rule we can see that:
1cos( ) sin( )
1sin( ) cos( )
ax b dx ax b ca
ax b dx ax b ca
Worked Example
sin(2 2) x dx 1cos(2 2)
2 x c
Integrating Trig Functions
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Higher Outcome 2
Integrating Trig Functions
Example
sin5 cos( 2 ) Evaluate t t dt
Apply the standard form
(twice).sin(5 ) s( 2 ) t dt co t dt 1 1
cos5 sin 25 2
t t c
Break up into two easier integrals
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Higher Outcome 2Example 2
4
3cos 22
Evaluate x dx
3cos 22
x dx( ) F x 3sin 2 x dx
Re-arrange 3
cos2 ( )2
x c
Apply the standard form
2
4
3cos 22
x dx
2 4
F F3
cos 2 cos 22 2 4
31 0
2
2
4
33cos 2
2 2
x dx
Integrating Trig Functions
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Higher Outcome 2
Integrating Trig Functions (Area)
Example
cosy xsiny x
1
x
y
1
0
AThe diagram shows the
graphs of y = -sin x and y = cos x
a) Find the coordinates of A
b) Hence find the shaded area cos sin The curves intersect where x x
tan 1 x
0 We want x
1tan ( 1) x C
AS
T0o180
o
270o
90o
3
2
2
3 7
4 4
and
3
4
x
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Higher Outcome 2
Integrating Trig Functions (Area)
( ) The shaded area is given by top curve bottom curve dx
3
4
0
(cos ( sin ))
Area x x dx
3
4
0
(cos sin )
x x dx
( ) F x (cos sin ) x x dx sin cos ( ) x x c
3
4
0
(cos sin )
Area x x dx3
(0)4
F F
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Higher Outcome 2
Integrating Trig Functions (Area)
3(0)
4
F F
3 3sin cos sin 0 cos0
4 4
sin cos 0 14 4
1 11
2 2
1 2 Area
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Higher Outcome 2Example 3cos3 cos(2 ) cos3 4cos 3cos a) By writing as show that x x x x x x
3cos b) Hence find x
cos(2 ) cos2 cos sin 2 sin x x x x x x
2 2(cos sin )cos (2sin cos )sin x x x x x x
2 2 2(cos sin )cos 2sin cos x x x x x
3 2cos 3sin cos x x x
3 2cos 3 1 cos cos x x x
3cos3 4cos 3cos x x x
Integrating Trig Functions
Remember cos(x +
y) =
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Higher Outcome 2
3cos3 4cos 3cos As shown above x x x
3 1cos (cos3 3cos )
4 x x x
3 1cos (cos3 3cos )
4 x dx x x dx
1(cos3 3cos )
4 x x dx
1 1 1(cos3 3cos ) sin3 3sin
4 4 3 x x dx x x c
3 1cos sin3 9sin
12 x dx x x c
Integrating Trig Functions
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Higher Outcome 2Example
5( ) '( ) 5 3 (1) 32 Find given and f x f x x f
5( ) 5 3 f x x dx
Use the “reverse” chain rule
5 15 3
5 5 1
xc 6
5 3
30
xc
So we have: 6
5 1 3(1) 32
30
f c
62
30 c
64
30 c
Giving:
32
15 c
32 1432 32
15 15 c 6
5 3 896( )
30 30
xf x
Integrating Functions
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Differentiation
Higher Mathematics
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Next
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Calculus Revision
Back NextQuit
Differentiate 24 3 7x x
8 3x
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Calculus Revision
Back NextQuit
Differentiate
3 26 3 9x x x
x
3 26 3 9x x x
x x x x
2 16 3 9x x x
Split up
Straight line form
22 6 9x x Differentiate
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Calculus Revision
Back NextQuit
Differentiate
3 1
2 22 5x x
1 3
2 23 12 2
2 5x x
1 3
2 25
23x x
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Calculus Revision
Back NextQuit
Differentiate2
3 1xx
1 1
2 23 2 1x x
Straight line form
Differentiate
1 3
2 21
2
32
2x x
1 3
2 23
2x x
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Calculus Revision
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Differentiate (3 5)( 2)x x
23 6 5 10x x x Multiply out
Differentiate 6 1x
23 10x x
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Calculus Revision
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Differentiate 2 8003 r
r
2 13 800r r Straight line form
Differentiate26 ( 1)800r r
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Calculus Revision
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Differentiate2( 2 )x x x
multiply out3 22x x
differentiate23 4x x
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Calculus Revision
Back NextQuit
Differentiate3
1 1
x x
1 3x x Straight line form
Differentiate2 4( 3)x x
2 43x x
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Calculus Revision
Back NextQuit
Differentiate 2x x x
1
22x x xStraight line form
multiply out
Differentiate
5 3
2 2x x3 1
2 25 3
2 2x x
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Calculus Revision
Back NextQuit
Differentiate 2 4x x
multiply out
Simplify
4 8 2x x x
Differentiate
6 8x x
Straight line form1
26 8x x 1
23 1x
![Page 44: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/44.jpg)
Calculus Revision
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Differentiate5( 2)x
Chain rule45( 2) 1x
45( 2)x Simplify
![Page 45: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/45.jpg)
Calculus Revision
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Differentiate3(5 1)x
Chain Rule23(5 1) 5x
215(5 1)x Simplify
![Page 46: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/46.jpg)
Calculus Revision
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Differentiate2 4(5 3 2)x x
Chain Rule 2 34(5 3 2) 10 3x x x
![Page 47: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/47.jpg)
Calculus Revision
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Differentiate
5
2(7 1)x
Chain Rule
Simplify
3
25
2(7 1) 7x
3
235
2(7 1)x
![Page 48: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/48.jpg)
Calculus Revision
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Differentiate
1
2(2 5)x
Chain Rule
Simplify
3
21
2(2 5) 2x
3
2(2 5)x
![Page 49: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/49.jpg)
Calculus Revision
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Differentiate 3 1x
Chain Rule
Simplify
Straight line form 1
23 1x
1
21
23 1 3x
1
23
23 1x
![Page 50: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/50.jpg)
Calculus Revision
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Differentiate 42
5
3x
Chain Rule
Simplify
Straight line form2 45( 3)x
2 520( 3) 2x x
2 540 ( 3)x x
![Page 51: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/51.jpg)
Calculus Revision
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Differentiate1
2 1x
Chain Rule
Simplify
Straight line form1
2(2 1)x
3
21
2(2 1) 2x
3
2(2 1)x
![Page 52: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/52.jpg)
Calculus Revision
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Differentiate2
2cos sin3
x x
22sin cos
3x x
![Page 53: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/53.jpg)
Calculus Revision
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Differentiate 3cos x
3sin x
![Page 54: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/54.jpg)
Calculus Revision
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Differentiate cos 4 2sin 2x x
4sin 4 4cos 2x x
![Page 55: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/55.jpg)
Calculus Revision
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Differentiate 4cos x
34cos sinx x
![Page 56: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/56.jpg)
Calculus Revision
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Differentiate sin x
Chain Rule
Simplify
Straight line form
1
2(sin )x1
21
2(sin ) cosx x
1
21
2cos (sin )x x
1
2
cos
sin
x
x
![Page 57: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/57.jpg)
Calculus Revision
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Differentiate sin(3 2 )x
Chain Rule
Simplify
cos(3 2 ) ( 2)x
2cos(3 2 )x
![Page 58: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/58.jpg)
Calculus Revision
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Differentiatecos5
2
x
Chain Rule
Simplify
1
2cos5xStraight line form
1
2sin 5 5x
5
2sin 5x
![Page 59: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/59.jpg)
Calculus Revision
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Differentiate ( ) cos(2 ) 3sin(4 )f x x x
( ) 2sin(2 ) 12cos(4 )f x x x
![Page 60: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/60.jpg)
Calculus Revision
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Differentiate 2 43200( )A x x
x
Chain Rule
Simplify
Straight line form2 1( ) 43200A x x x
2( ) 2 43200A x x x
2
43200( ) 2A x x
x
![Page 61: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/61.jpg)
Calculus Revision
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Differentiate 2
2( )f x x
x
Differentiate
Straight line form1
22( ) 2f x x x
1321
2( ) 4f x x x
![Page 62: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/62.jpg)
Calculus Revision
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Differentiate 3 22 7 4 4y x x x
26 14 4y x x
![Page 63: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/63.jpg)
Calculus Revision
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Differentiate 62siny x
Chain Rule
Simplify
62 cos 1
dyx
dx
62 cos
dyx
dx
![Page 64: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/64.jpg)
Calculus Revision
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Differentiate3
(8 )4
A a a
multiply out
Differentiate
236
4A a a
36
2A a
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Calculus Revision
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Differentiate
13 2( ) (8 )f x x
Chain Rule
Simplify
13 221
2( ) (8 ) ( 2 )f x x x
12 3 2( ) (8 )f x x x
![Page 66: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/66.jpg)
Calculus Revision
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Differentiate16
, 0y x xx
Differentiate
Straight line form1
216y x x
3
21 8dy
xdx
![Page 67: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/67.jpg)
Calculus Revision
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Differentiate 23 3 16( )
2A x x
x
Straight line form
Multiply out 23 3 3 3 16( )
2 2A x x
x
Differentiate
2 13 3( ) 24 3
2A x x x
2( ) 3 3 24 3A x x x
![Page 68: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/68.jpg)
Calculus Revision
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Differentiate 1
2( ) 5 4f x x
Chain Rule
Simplify
1
21
2( ) 5 4 5f x x
1
25
2( ) 5 4f x x
![Page 69: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/69.jpg)
Calculus Revision
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Differentiate 216( ) 240
3A x x x
32( ) 240
3A x x
![Page 70: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/70.jpg)
Calculus Revision
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Differentiate 2( ) 3 (2 1)f x x x
Multiply out
Differentiate
3 2( ) 6 3f x x x
2( ) 18 6f x x x
![Page 71: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/71.jpg)
Calculus Revision
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Differentiate2 2( ) cos sinf x x x
( ) 2 cos sin 2sin cosf x x x x x Chain Rule
Simplify ( ) 4 cos sinf x x x
![Page 72: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/72.jpg)
Quit
C P Dwww.maths4scotland.co.uk
© CPD 2004
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Integration
Higher Mathematics
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Next
![Page 74: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/74.jpg)
Calculus Revision
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Integrate 2 4 3x x dx 3 24
33 2
x xx c
3 212 3
3x x x c
Integrate term by term
simplify
![Page 75: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/75.jpg)
Calculus Revision
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Find 3cos x dx3sin x c
![Page 76: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/76.jpg)
Calculus Revision
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Integrate (3 1)( 5)x x dx 23 14 5x x dx
3 23 145
3 2
x xx c
3 27 5x x x c
Multiply out brackets
Integrate term by term
simplify
![Page 77: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/77.jpg)
Calculus Revision
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Find 2sin d 2cos c
![Page 78: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/78.jpg)
Calculus Revision
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Integrate2(5 3 )x dx
3(5 3 )
3 3
xc
31
9(5 3 )x c
Standard Integral
(from Chain Rule)
![Page 79: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/79.jpg)
Calculus Revision
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Find p, given
1
42p
x dx 1
2
1
42p
x dx 3
2
1
2
342
p
x
3 3
2 22 2
3 3(1) 42p
2 233 3
42p 32 2 126p
3 64p 3 2 1264 2p 1
12 32 16p
![Page 80: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/80.jpg)
Calculus Revision
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Evaluate
2
21
dx
x2
2
1
x dx 21
1x 1 12 1
11
2
1
2
Straight line form
![Page 81: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/81.jpg)
Calculus Revision
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Find
1
6
0
(2 3)x dx17
0
(2 3)
7 2
x
7 7(2 3) (0 3)
14 14
7 75 3
14 14
5580.36 156.21 (4sf)5424
Use standard Integral
(from chain rule)
![Page 82: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/82.jpg)
Calculus Revision
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Find1
3sin cos2
x x dx1
3cos sin2
x x c Integrate term by term
![Page 83: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/83.jpg)
Calculus Revision
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Integrate3
2x dxx
1
32 2x x dx 3 222
3
2
2
xx c
3222
3x x c
Straight line form
![Page 84: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/84.jpg)
Calculus Revision
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Integrate3 1x dx
x
13 2x x dx
14 2
1
24
x xc
14 21
42x x c
Straight line form
![Page 85: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/85.jpg)
Calculus Revision
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Integrate
3 5x xdx
x
1 1
2 2
3 5x xdx
x x
5 1
2 25x x dx 7 3
2 22
7
10
3cx x
Straight line form
![Page 86: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/86.jpg)
Calculus Revision
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Integrate
324
2
x xdx
x
3
2
1 1
2 2
4
2 2
x xdx
x x
1
2 1
22x x dx
3
2 24 1
3 4x x c
Split into separate fractions
![Page 87: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/87.jpg)
Calculus Revision
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Find 2
3
1
2 1x dx
24
1
2 1
4 2
x
4 44 1 2 1
4 2 4 2
4 45 3
8 8
68
Use standard Integral
(from chain rule)
![Page 88: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/88.jpg)
Calculus Revision
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Find sin 2 cos 34
x x dx
1 1
2 3cos 2 sin 3
4x x c
![Page 89: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/89.jpg)
Calculus Revision
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Find2
sin7
t dt2
7cos t c
![Page 90: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/90.jpg)
Calculus Revision
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Integrate
1
2
1
0 3 1
dx
x
1
2
1
0
3 1x dx
Straight line form
1
2
1
0
13
2
3 1x
1
0
23 1
3x
2 23 1 0 1
3 3
2 24 1
3 3
4 2
3 3
2
3
![Page 91: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/91.jpg)
Calculus Revision
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Given the acceleration a is: 1
22(4 ) , 0 4a t t
If it starts at rest, find an expression for the velocity v where dv
adt
1
22(4 )dv
tdt
3
2
31
2
2(4 )tv c
3
24
(4 )3
v t c
344
3v t c Starts at rest, so
v = 0, when t = 0 34
0 43
c
340 4
3c
320
3c
32
3c
3
24 32
3 3(4 )v t
![Page 92: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/92.jpg)
Calculus Revision
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A curve for which 3sin(2 )dy
xdx
5
12, 3passes through the point
Find y in terms of x. 3
2cos(2 )y x c
3 5
2 123 cos(2 ) c Use the point 5
12, 3
3 5
2 63 cos( ) c 3 3
2 23 c
3 33
4c
4 3 3 3
4 4c
3
4c
3 3
2 4cos(2 )y x
![Page 93: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/93.jpg)
Calculus Revision
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Integrate 2 2
2
2 2, 0
x xdx x
x
4
2
4xdx
x
4
2 2
4xdx
x x 2 24x x dx
3 14
3 1
x xc
31
3
4x c
x
Split into
separate fractions
Multiply out brackets
![Page 94: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/94.jpg)
Calculus Revision
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If
( ) sin(3 )f x x passes through the point 9, 1
( )y f x express y in terms of x.
1
3( ) cos(3 )f x x c Use the point 9
, 1
1
31 cos 3
9c
1
31 cos
3c
1 1
3 21 c
7
6c 1 7
3 6cos(3 )y x
![Page 95: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/95.jpg)
Calculus Revision
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Integrate2
1
(7 3 )dx
x2(7 3 )x dx
1(7 3 )
1 3
xc
11
3(7 3 )x c
Straight line form
![Page 96: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/96.jpg)
Calculus Revision
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The graph of
32
1 1
4
dyx
dx x
( )y g x passes through the point (1, 2).
express y in terms of x. If
3 2 1
4
dyx x
dx
4 1 1
4 1 4
x xy x c
simplify
4 1 1
4 4
xy x c
x Use the point
41 1 1
2 14 1 4
c
3c Evaluate c41 1
4 4
13y x x
x
![Page 97: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/97.jpg)
Calculus Revision
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Integrate2 5x
dxx x
3
2
2 5xdx
x
3 3
2 2
2 5xdx
x x
1 3
2 25x x dx
3 1
2 2
3 1
2 2
5x xc
3 1
2 22
310x x c
Straight line form
![Page 98: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/98.jpg)
Calculus Revision
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A curve for which 26 2dy
x xdx
passes through the point (–1, 2).
Express y in terms of x.
3 26 2
3 2
x xy c 3 22y x x c
Use the point3 22 2( 1) ( 1) c 5c
3 22 5y x x
![Page 99: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/99.jpg)
Calculus Revision
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Evaluate22
2
1
1x dxx
1
222
1x x dx
Cannot use standard integral
So multiply out
4 22
12x x x dx
25 2 1
1
1
5x x x 5 2 5 21 1 1 1
5 2 5 12 2 1 1
32 1 1
5 2 54 64 40 20 2
10 10 10 10 82
10 1
58
![Page 100: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/100.jpg)
Calculus Revision
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Evaluate4
1
x dx1
2
4
1
x dx 43
2
1
2
3x
4
1
32
3x
3 32 24 1
3 3
16 2
3 3
14
3
2
34
Straight line form
![Page 101: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Harder Type Questions Further Integration](https://reader036.vdocuments.site/reader036/viewer/2022062409/5697c0291a28abf838cd7425/html5/thumbnails/101.jpg)
Calculus Revision
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Evaluate0
2
3
(2 3)x dx
Use standard Integral
(from chain rule)
03
3
(2 3)
3 2
x
3 3(2(0) 3) (2( 3) 3)
6 6
27 27
6 6
27 27
6 6
54
6 9
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Calculus Revision
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The curve ( )y f x ,112
passes through the point
( ) cos 2f x x Find f(x)
1
2( ) sin 2f x x c
1
2 121 sin 2 c
use the given point ,112
1
2 61 sin c
1 1
2 21 c 3
4c 1 3
2 4( ) sin 2f x x
1
6 2sin
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Calculus Revision
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2(6 cos )x x x dx Integrate
3 26sin
3 2
x xx c Integrate term by term
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Calculus Revision
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Integrate 33 4x x dx4 23 4
4 2
x xc
4 23
42x x c
Integrate term by term
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Calculus Revision
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Evaluate1
01 3x dx
1
21
01 3x dx
13
2
0
3
2
1 3
3
x
13
2
0
21 3
9x
13
0
21 3
9x
3 32 21 3(1) 1 3(0)
9 9
3 32 24 1
9 9
16 2
9 9
14
9 5
91
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