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integrand② Definite Integral a

①f fcxildxtnoiaidshrmsovnn,q

dx-290×0Integral sign

o-

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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

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fan.ii. ±

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fan -a X

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"

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

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fan = a ⇒ FM = A- + c

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RX d

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∫0dx = C

kX d

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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

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x− 4

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x

�

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522 lol C← 513 - 4×454 t C , C = diorama

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z

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⑥ f 4dB = 4S + e

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te

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Ry9

1t�KTH2 8XR

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x5 + 2x3 − 1

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U2V∫(4 sinx+ 2 cosx)dx =

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f ÷×adx= arctanxtc

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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

fC2xtsP°d¥"hwdriving[ hi u -- 2×+5 ifrrisnds bed brooklime

⇒ !¥!d÷= . ⇒ an . edx ⇒

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37= tag + c

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t C

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Irfan U -- 4×+8 f fondue

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J cos u. If = I feosudn = I

,

Shilton

= I,

sin ( 4×+87 dx

Q :Eo#s?t

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RX∫

du = u+ C

kX∫

undu =un+1

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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

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du

a2 + u2=

1

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u

a) + C

Ry3

it. uex,f d D= D te

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abosun

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