integral sign i.math.science.cmu.ac.th/thaned/cal111/docs1-63/lec15-111.pdf · 2020. 9. 14. ·...
TRANSCRIPT
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integrand② Definite Integral a
①f fcxildxtnoiaidshrmsovnn,q
dx-290×0Integral sign
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1-7-
ffcxidx = ffct> dtin
a.imr8a !fun x' fat, at
f fix ,d④ = ffchdt1
Hoax in = ffczydzzDummy
variable= f. fundus
( dy )& ffcs> DE
9dm anti derivative ro. fan n'ohhh
① fan = x ⇒ Fan -- ?
run off, a fan = x#× Fan = x
'⇒ check : dat; - dy = {x
" ⇒ a- ÷:÷÷¥¥:*.
Irfan text = If -15 ✓
Moby : off,= da,( Ees) = If to = X - fat
ioan Fam -
- Ia - I, ✓
Eoe doff,
= If -- x -
ca arbitrary constant
=xItcedan¥ ri::÷IIime.. ..
fans xd ⇒ Fan - ?
FCK = XI + C4
- 5 - 4
fan = X ⇒ Fix = I + C-4
E@ @ fan-
- X
"
⇒ Fan = I" + cNtl
Ntl =/
fan.ii. ±
° ⇒
@ Fcxt⇒ DIE - ¥nx = ¥
fan -a X
"
, Uf - I ⇒ Fan = x÷,
"
t c
fix, = x"
⇒ Fan = ln 1×1 t C
fan - Smx ⇒ Fan = - cosx + e
few = Cos X =) Fan = Sihx t C
-
fan = faux ⇒ Fan = ?
fan = cotx ⇒ Fix , = ?
fan = sax ⇒ ?
fan e ooseex e) ?-
fcxi = seek =) Fan = tanx + c
fan =see x. tan 'x ⇒ Fan = see x + c
fan = coset x ⇒ Fan = - cotx + c
fan = coseex - atx ⇒ fan = - cosec x + c
-
X
fan a e =) Fan = EX t Cx x
fan = a ⇒ FM = A- + c
haax
da, a
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Ryj
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3x3,
1
3x3 + 1,
1
3x3 − 3,
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3x3 −
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RX d
dx(C) = 0 RX
∫0dx = C
kX d
dx[kx] = k kX
∫kdx = kx+ C
jX d
dx[kf(x)] = kf ′(x) jX
∫[kf(x)]dx = k
∫f(x)dx
9X d
dx[f(x)± g(x)] = f ′(x)± g′(x) 9X
∫[f(x)± g(x)]dx =
∫f(x)dx±
∫g(x)dx
8X d
dx[xn] = nxn−1 8X
∫xndx =
xn+1
n+ 1+ C, n #= −1
eX d
dx[ln |x|] = 1
xeX
∫1
xdx = ln |x|+ C
dX d
dx[ex] = ex dX
∫exdx = ex + C
3X d
dx[ax] = ax ln a, a > 0 �M/ a #= 1 3X
∫axdx =
ax
ln a+ C, a > 0 �M/ a #= 1
NX d
dx[sinx] = cosx NX
∫cosxdx = sinx+ C
RyX d
dx[cosx] = − sinx RyX
∫sinxdx = − cosx+ C
RRX d
dx[tanx] = sec2 x RRX
∫sec2 xdx = tanx+ C
RkX d
dx[cotx] = −cosec2x RkX
∫cosec2xdx = − cotx+ C
RjX d
dx[secx] = secx tanx RjX
∫secx tanxdx = secx+ C
R9X d
dx[cosecx] = −cosecx cotx R9X
∫cosecx cotxdx = −cosecx+ C
R8X d
dx[arcsinx] =
1√1− x2
R8X∫
1√1− x2
dx = arcsinx+ C
ReX d
dx[arctanx] =
1
1 + x2ReX
∫1
1 + x2dx = arctanx+ C
RdX d
dx[ln | secx|] = tanx RdX
∫tanxdx = ln | secx|+ C
R3X d
dx[ln | sinx|] = cotx R3X
∫cotxdx = ln | sinx|+ C
kyeRRR, *�H+mHmb R �+�/2KB+ v2�` kyky
f fun -gun dx = ?
Hg¥, dx -e ?HogarTyr,w
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��∫(5x2 − 8x+ 5) dx
��∫(x3/2 + 2x+ 3) dx
��∫(√x+
1
3√x) dx
��∫
x2 + 4
xdx
��∫(1 + 3t)t2 dt
��∫
4 ds
��∫
y2 4√y dy
��∫
5 VLQx dx
��∫(6x3 + 9x2 + 4x− 7) dx
���∫ (
4
x− 4
x2
)dx
���∫
3x5/6 − 9x4/3 dx
���∫
1
x√xdx
���∫
4ex − FRVx dx
���∫
1
3(x+
4√x5) dx
��®µ F (x) �̧É°��¨o°���́ F (x) =∫f(x)dx Á¤ºÉ°
��� f(x) = 40
��� f(x) = 3x4
��� f(x) = 5ex
��� f(x) =2
x
�
fan = 40 ⇒ Fox) -- Gox + c
④ fall = 3×4 ⇒ Fox) = 3¥! C⑦ fan = 5e× ⇒ Fan = b-exec
④ fix, = Ex ⇒ Fix = 2/41×1 te
① J (5×2-8×+5) dx
= fsxtdx - fsxdx t fsdx ~f×④= 5) x'dx - 8) xdx c- 5 fdx= @ II t Ci ) - 8C + ca ) + (5×+5)= (53×3 - 8¥ t 5x ) t Kita -eat Cg )
522 lol C← 513 - 4×454 t C , C = diorama
-
@ {(x"'t ex + s ) dx5/3
= ¥,
t II t 3 X t C #
@ f×dx= f xktzxkdx
312 k= I e 3¥ t CHu
z
④ f KII dx - fifty,
dx
= fix t ¥ dx = Eat 4.lu/xt- c-
⑤ f @ eat' ) tdt= f @ t + et
' )dt = E'+ ¥t4e-
⑥ f 4dB = 4S + e
-
C f du = at c
⑧ f Ze Ixzdx = 3h44 + 4¥'
te
⑨ fxtpdx - fi"dx =
+ a- to
④ ffxtsiux)dx = II - cos x * e
Ry9
1t�KTH2 8XR
U�V∫(x6 − 7x+ 4)dx =
U#V∫
x5 + 2x3 − 1
x4dx =
U+V∫(√x+
13√x)dx =
U/V∫(ex + 2x)dx =
U2V∫(4 sinx+ 2 cosx)dx =
U7V (3√
1− x2− 2
1 + x2)dx =
�+�/2KB+ v2�` kyky kyeRRR, *�H+mHmb R
7-
If - 7¥ -14xtc
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t C-3
3/2 43I t I t C
3/2 43
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th2
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J § are Sm ex) - 2 arc tan Cx) + c
J dx - arcsmx e c
f ÷×adx= arctanxtc
Sexts) dx = 2zX2t5X t c
J @x-is ) x dx e f 2×2 t5X DX - 23¥ t 4C§ Cut55 dx = 14×2-1 text 25 ) dy
z 4133 t 2oxt 25X t C
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t C
J cos ( 4×-18 )dx = ?
Irfan U -- 4×+8 f fondue
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J cos u. If = I feosudn = I
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Shilton
= I,
sin ( 4×+87 dx
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RX∫
du = u+ C
kX∫
undu =un+1
n+ 1+ C, n #= −1
jX∫
1
udu = ln |u|+ C
9X∫
au du =au
ln a+ C, a > 0, a #= 1
8X∫
eu du = eu + C
eX∫
sinu du = − cosu+ C
dX∫
cosu du = sinu+ C
3X∫
sec2u du = tanu+ C
NX∫
csc2u du = − cotu+ C
RyX∫
sec u tanu du = secu+ C
RRX∫
csc u cotu du = − cscu+ C
RkX∫
tanu du = ln| secu|+ C
RjX∫
cotu du = ln| sinu|+ C
R9X∫
du√a2 − u2
= arcsin(u
a) + C
R8X∫
du
a2 + u2=
1
aarctan(
u
a) + C
Ry3
it. uex,f d D= D te
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