Understanding Engineering Mathematics

9.3.11 Choice of integration methods

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A.From the following three methods of integration:

A standard integral

B substitution

C integration by parts

state which you would use to evaluate the following integrals:

(i)

∫

2 exdx (ii)

∫

xexdx (iii)

∫

xex

2

dx

(iv)

∫

xcos(x^2 )dx (v)

∫

secxtanxdx (vi)

∫

sec^2 xtanxdx

B. Choose from

A partial fractions

B completing the square

C substitution

the methods (in the order in which you would use them) you would use to integrate

(i)

1

x^2 +x+ 1

(ii)

1

x^2 +x− 2

(iii)

2 x+ 1

x^2 +x− 1

(iv)

4 x+ 12

√

x^2 + 6 x+ 6

(v)

1

√

7 − 6 x−x^2

(vi)

2

x^2 + 2 x+ 1

C.Integrate the integrals in Review Question 9.1.11.

9.3.12 The definite integral

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A.Evaluate

(i)

∫ 1

0

x^2 dx (ii)

∫π/ 2

0

xcosxdx (iii)

∫ 1

0

x^2

√

x^3 + 1

dx

(iv)

∫ 1

0

x

(x+ 1 )(x+ 2 )

dx (v)

∫ 1

0

xexdx

B.Iff(x)is an even function andg(x)is an odd function, show that

(i)

∫a

−a

f(x)dx= 2

∫a

0

f(x)dx (ii)

∫a

−a

g(x)dx= 0