Understanding Engineering Mathematics

(v) sin 3x (vi) cosx^3 (vii) e^5 x (viii)

1

x+ 1

(ix) ln 2x (x)

√

x+1(xi)ln( 3 x+ 1 ) (xii) tan−^1 x

B. Integrate the following functions:

(i)

2

x

(ii) 3e^5 x (iii)

2

( 1 +x)^2

(iv)

1

x^5

(v) 3x^2 cosx^3 (vi)

− 3

x^2 + 1

(vii) x−

3

(^5) (viii)
1
√
x
(ix) 2 cos 3x (x) x^2 (xi)
3
√
x+ 1
(xii)
1
3 x+ 1
9.3.2 Standard integrals
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A.Integrate the following functions:
(i) 4 (ii) 3x^2 − 2 x+ 1 (iii) 3/x^5 (iv) x^2 /^3
(v) cosx (vi) sec^2 x (vii) lnx (viii)
1
x^2 + 1
B. Integrate the following functions with respect to the appropriate variables:
(i) 4u (ii) 2s^2 − 3 s+ 2 (iii) 6/t^7 (iv) x^1 /^4
(v) sinθ (vi) cosec^2 t (vii) e^3 t (viii)
1
s^2 + 1
9.3.3 Addition of integrals
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➤
Integrate:
(i) 2x^2 −
3
x^3
(ii) 2 cosx−sinx (iii) coshx=
1
2
(ex+ex)
(iv) sin(x+ 3 )by compound angle formulae
(v) The polynomial:
∑n
r= 0
arxr
(vi) sinxcos 2x (vii) sinhx
9.3.4 Simplifying the integrand
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➤
A.Evaluate the following by simplifying the integrand – noting any special conditions
needed.