name the keyword or the method. keywords: life-saving principle i and the natural log function
TRANSCRIPT
NAME THE KEYWORD
OR THE METHOD
dx
x
x
42 dx
x 4
12
dxx 4
12
dx
x 4
12
dx
x 24
1
dxx
x24
dxx
x
42
2
dx
x
x
42
2
dxx
x
42
2
dx
x 4
12
dx
x
x
42
3
dx
xx 4
12
dxx
x
42 dx
x
x
4
122
dx
x
x
4
122
dx
x
x
4
122
dxx
x32 )4(
dxx 32 )4(
1 dxx 24 dxxx 24
dxx 42 dxxx 422
dxxx
x
45
322
dxx
x
83
2
dx
x
x
83
dx
x
x
42
Keywords: Life-Saving Principle I and the natural log function
Cxdxx
x
|4|ln2
1
42
2
dx
x 4
12
Keywords: Inverse tangent function
Cx
Tandxx
22
1
4
1 12
Tanx 2
dxx 4
12
Keywords: Trig substitution leading into the natural log function, or just use the standard formula.
Cxxdxx
|4|ln4
1 2
2
dx
x 4
12
Cxxdxx
|4|ln4
1 2
2
Keywords: Trig substitution Tanx 2leading into the natural log function, or just use the standard formula.
dxx 24
1
Cx
Sindxx
24
1 1
2
Keywords: Inverse sine function
Keywords: u-substitution .The trig substitution is not necessary, but still will work.
dxx
x24
24 xu
Cxdxx
x
2
24
4
Sinx 2
dx
x
x
42
2
dxx
dxx
xdx
x
x
4
41
4
4)4(
4 22
2
2
2
Keywords: Long Division and inverse tangent
Cx
Tanx
22 1
dx
x
x
42
2
Secx 2
dTanSecdx
Secx
2
2
Keywords: Trig substitution
TanTanSecx 24444 222
dSecdTanSecTan
Secdx
x
x 32
2
2
422
4
4
Now use Integration by Parts and continue
dxx
x
42
2
Tanx 2
dTanSecdSecSec
Tandx
x
x 222
2
2
422
4
4
Keywords: Trig substitution
dSecdx
Tanx22
2
SecSecTanx 24444 222
||ln4)1(4 32 TanSecdSecdSecSec
Now use Integration by Parts on the remaining integral
dx
x 4
12
Keywords: Partial Fractions! This is because the denominator breaks into factors.
dx
x
x
42
3
Keywords: Long division followed by Life-Saving Principle I. Partial Fractions is really not be needed.
dx
xx 4
12
Keywords: Completing the square, and the inverse tangent.
Cx
Tandx
x
dxxx
15
12
15
2
4
15
2
1
1
4
1 122
dx
x
x
42
Keywords: Life-Saving Principle I, leading to the natural logarithm. Partial Fractions are not needed!
Cxdxx
x
|4|ln2
1
42
2
dxx
x
4
122
Keywords: Partial Fractions
C
xTanxdx
xdx
x
xdx
x
x
22
1|4|ln
4
1
4
2
4
12 12222
Keywords: Break the integral. Then use Life-Saving Principle I and inverse tangent
dxx
x
4
122
dx
x
x
4
122
Cxxx |4|ln42 22
Keywords: Break the integral. Then use u-substitution (or candidate method) and a trig substitution (or the standard formula with natural log function)
4
1
4
2
4
12222 x
dxx
xdx
x
x
dx
x
x32 )4(
Cx
dxx
x
2232 )4(4
1
)4(
Keywords: Use a u-substitution or the candidate function method.
42 xu
Keywords: Use the trig substitution . There is no radical term, but still this substitution really works!
dx
x 32 )4(
1
Tanx 2
dCosdSecSec
dxx
42632 32
12
64
1
)4(
1
dSecdx
Tanx22
2
222 444)4( SecTanx
Now continue using one of the Double Angle formulas for the cosine
Keywords: Use the trig substitution
dxx 24
Sinx 2
dCosdCosCosdxx 22 42.24
dCosdx
Sinx
2
2
CCosSinC
Sind
Cos)(2
2
22
2
)21(4
Cxxx
Sin
4
4
22
21
CosCosSinx 24444 222
Keywords: Use a u-substitution or the candidate function method. The trig substitution is not necessary.
dxxx 24
24 xu
Cxdxxx 2/322 )4(3
14
Sinx 2
Keywords: Use the trig substitution
dxx 42
Tanx 2
dSecdx
Tanx22
2
SecSecTanx 24444 222
dSecdSecSecdxx 322 42.24
Now use Integration by parts and continue.......
Keywords: Use the trig substitution
dxxx 422
Secx 2
dTanSecdTanSecTanSecdxxx 23222 162.2.44
dTanSecdx
Secx
2
2
TanTanSecx 24444 222
Now use Integration by parts and continue.......
dxxx
x
45
322
Keywords: Use Partial Fractions. Since the denominator breaks into factors, this method is better than attempting to complete the square.
dx
x
x
83
2
Keywords: Life-Saving Principle I, leading to the natural logarithm function. Partial Fractions are not needed, even if the denominator breaks into factors!
Cxdxx
x
|8|ln3
1
83
3
2
dx
x
x
83
Keywords: Partial Fractions are needed!
IT IS CRUCIALTO MEMORIZE THE
FOLLOWING FIVE
FORMULAS....
MEMORIZE!
Ca
xSindx
xa
1
22
1
Ca
xTana
dxxa
122
11
21
122
22||ln
1C
a
xSinhCaxxdx
ax
21
122
22||ln
1C
a
xCoshCaxxdx
ax
Ca
xSecdx
axx
1
22||
1
LIFE-SAVING PRINCIPLE I
Cxfdxxf
xf |)(|ln
)(
)('