Understanding Engineering Mathematics

9.3.7 Integrating rational functions

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A.Integrate the following by partial fractions:

(i)

2 x

(x− 1 )(x+ 3 )

(ii)

x+ 1

x^2 + 5 x+ 6

(iii)

4

2 x^2 −x− 1

(iv)

3

(x+ 1 )(x^2 + 1 )

(v)

x+ 1

(x− 1 )^2 (x− 2 )

(vi)

2 x+ 1

x^3 + 2 x^2 −x− 2

B. Integrate the following by completing the square:

(i)

1

x^2 + 2 x+ 5

(ii)

3

x^2 − 2 x+ 2

(iii)

2

2 x^2 + 2 x+ 1

(iv)

1

x^2 + 6 x+ 10

(v)

1

2 x^2 + 12 x+ 27

C.Integrate:

(i)

x^2 + 1

(x+ 1 )(x+ 2 )

(ii)

x^3

x^2 + 2 x+ 2

(iii)

3 x^4

(x− 1 )(x+ 1 )

9.3.8 Using trig identities in integration

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Integrate:

(i) cos^2 xsin^3 x (ii) cos 2xcos 3x (iii) cos^5 x

(iv) cos 5xsin 3x (v) sin 2xsin 3x

9.3.9 Using trig substitutions in integration

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A.Integrate the following functions using appropriate substitutions:

(i)

1

√

4 − 4 x^2

(ii)

2

4 + 9 x^2

(iii)

2

√

1 − 9 x^2

(iv)

3

1 + 4 x^2

(v)

1

√

8 − 2 x−x^2

(vi)

1

√

6 x−x^2

B.Use thet=tan

θ

2

substitution to integrate

∫

1

3 +5cosθ

dθ.

9.3.10 Integration by parts

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Integrate:

(i) xcosx (ii) x^3 ex (iii) sin−^1 x (iv) excosx

(v) x^2 cosx (vi) xlnx (vii) x^3 ex

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