Understanding Engineering Mathematics

9.3.7 Integrating rational functions

A. (i)

1

2

ln|(x− 1 )(x+ 3 )^3 | (ii) ln

∣

∣

∣

∣

(x+ 3 )^2

(x+ 2 )

∣

∣

∣

∣ (iii)

4

3

ln

∣

∣

∣

∣

x− 1

2 x+ 1

∣

∣

∣

∣

(iv)

3

4

ln

∣

∣

∣

∣

(x+ 1 )^2

x^2 + 1

∣

∣

∣

∣+

3

2

tan−^1 x (v) 3 ln

∣

∣

∣

∣

x− 2

x− 1

∣

∣

∣

∣+

2

x− 1

(vi)

1

2

ln

∣

∣

∣

∣

x^2 − 1

(x+ 2 )^2

∣

∣

∣

∣

B. (i)

1

2

tan−^1

(

x+ 1

2

)

(ii) 3 tan−^1 (x− 1 ) (iii) 2 tan−^1 ( 2 x+ 1 )

(iv) tan−^1 (x+ 3 ) (v)

1

3

√

2

tan−^1

[√

2 (x+ 3 )

3

]

C. (i) x+

1

3

ln

∣

∣

∣

∣

(x− 2 )^5

(x+ 1 )^2

∣

∣

∣

∣ (ii)

x^2

2

− 2 x+ln(x^2 + 2 x+ 2 )+2tan−^1 (x+ 1 )

(iii) x^3 + 3 x+

3

2

ln

∣

∣

∣

∣

x− 1

x+ 1

∣

∣

∣

∣

9.3.8 Using trig identities in integration

(i)

cos^5 x

5

−

cos^3 x

3

(ii)

1

10

sin 5x+

1

2

sinx

(iii) sinx−

2

3

sin^3 x+

1

5

sin^5 x (iv) −

1

16

(cos 8x−4cos2x)

(v)

1

10

(5sinx−sin 5x)

9.3.9 Using trig substitutions in integration

A. (i)

1

2

sin−^1 x (ii)

1

3

tan−^1

(

3 x

2

)

(iii)

2

3

sin−^13 x

(iv)

3

2

tan−^12 x (v) sin−^1

(

x+ 1

3

)

(vi) sin−^1

(

x− 3

3

)

B.

1

4

ln

∣

∣

∣

∣

2 +tanθ/ 2

2 −tanθ/ 2

∣

∣

∣

∣

9.3.10 Integration by parts

(i) xsinx+cosx (ii) (x^3 − 3 x^2 + 6 x− 6 )ex (iii) xsin−^1 x+

√

1 −x^2