Understanding Engineering Mathematics

(i)

∫

x^2 − 3 x+ 2

x− 2

dx (ii)

∫

lnecosxdx

(iii)

∫

cos 2x

cosx+sinx

dx (iv)

∫

ecos

(^2) x
lne^2 x
e3lnxe−sin
(^2) xdx
(v)
∫
(cosx−sinx)^2 dx
B. Sometimes you have tocomplicatea function to integrate it. Integrate secxby multi-
plying and dividing by secx+tanx. Integrate cosecxin a similar way.
9.3.5 Linear substitution in integration
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Integrate:
(i) ( 4 x+ 3 )^7 (ii) ( 2 x− 1 )^1 /^3 (iii) sin( 3 x− 1 ) (iv) 2e^4 x+ 1
(v)
3
2 x+ 1
(vi) ( 2 x− 1 )^4 (vii)
√
3 x+ 2 (viii) cos( 2 x+ 1 )
(ix) 4e^3 x−^1 (x)
1
4 x− 3
9.3.6 Thedu=f′.x/dxsubstitution
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A.Write down the substitutionu=f(x)you would use to integrate the following and
hence integrate them.
(i) x^2 ex
3
(ii) sec^2 xsin(tanx+ 2 )
(iii) cosxsin^3 x (iv)
x+ 1
x^2 + 2 x+ 3
(v) tanx
B. Integrate:
(i) f′(x)sin(f (x)) (ii) f(x)e(f (x))
2
f′(x) (iii) x^2 cos
(
x^3 + 1
)
(iv) −sinxln(cosx) (v)
x− 1
x^2 − 2 x+ 1
(vi)
∫
f′(x)ef(x)dx
(vii)
∫
f′(x)cos(f (x)+ 2 )dx
C.Integrate:
(i) 3x^2 (x^3 + 2 )^3 (ii) 2x
√
x^2 + 1 (iii)
2 x− 1
x^2 −x+ 1
(iv) (x+ 1 )cos(x^2 + 2 x+ 1 ) (v) sinxecosx (vi)
sec^2 x
tanx
(vii) cos 2x(2sin2x+ 1 )^3 (viii)
6 x+ 12
√
x^2 + 4 x+ 4