Higher Engineering Mathematics, Sixth Edition

404 Higher Engineering Mathematics

4. Determine

∫ √

( 16 − 9 t^2 )dt.

[

8

3

sin−^1

3 t

4

+

t

2

√

( 16 − 9 t^2 )+c

]

5. Evaluate

∫ 4

0

1

√

( 16 −x^2 )

dx.

[π

2

or 1. 571

]

6. Evaluate

∫ 1

0

√

( 9 − 4 x^2 )dx. [2.760]

40.6 Worked problems on integration

usingtanθsubstitution

Problem 17. Determine

∫

1

(a^2 +x^2 )

dx.

Letx=atanθthen

dx

dθ

=asec^2 θand dx=asec^2 θdθ.

Hence

∫

1

(a^2 +x^2 )

dx

=

∫

1

(a^2 +a^2 tan^2 θ)

(asec^2 θdθ)

=

∫

asec^2 θdθ

a^2 ( 1 +tan^2 θ)

=

∫

asec^2 θdθ

a^2 sec^2 θ

,since1+tan^2 θ=sec^2 θ

=

∫

1

a

dθ=

1

a

(θ )+c

Sincex=atanθ,θ=tan−^1

x

a

Hence

∫

1

(a^2 +x^2 )

dx=

1

a

tan−^1

x

a

+c

Problem 18. Evaluate

∫ 2

0

1

( 4 +x^2 )

dx.

From Problem 17,

∫ 2

0

1

( 4 +x^2 )

dx

=

1

2

[

tan−^1

x

2

] 2

0

sincea= 2

=

1

2

(tan−^11 −tan−^10 )=

1

2

(π

4

− 0

)

=

π

8

or 0. 3927

Problem 19. Evaluate

∫ 1

0

5

( 3 + 2 x^2 )

dx, correct

to 4 decimal places.

∫ 1

0

5

( 3 + 2 x^2 )

dx=

∫ 1

0

5

2[( 3 / 2 )+x^2 ]

dx

=

5

2

∫ 1

0

1

[

√

( 3 / 2 )]^2 +x^2

dx

=

5

2

[

1

√

( 3 / 2 )

tan−^1

x

√

( 3 / 2 )

] 1

0

=

5

2

√(

2

3

)[

tan−^1

√(

2

3

)

−tan−^10

]

=( 2. 0412 )[0. 6847 −0]

= 1. 3976 , correct to 4 decimal places.

Now try the following exercise

Exercise 159 Further problems on

integration using thetanθsubstitution

1. Determine

∫

3

4 +t^2

dt.

[

3

2

tan−^1

t

2

+c

]

2. Determine

∫

5

16 + 9 θ^2

dθ.

[

5

12

tan−^1

3 θ

4

+c

]

3. Evaluate

∫ 1

0

3

1 +t^2

dt. [2.356]

4. Evaluate

∫ 3

0

5

4 +x^2

dx. [2.457]

40.7 Worked problems on integration

using thesinhθsubstitution

Problem 20. Determine

∫

1

√

(x^2 +a^2 )

dx.

Letx=asinhθ,then

dx

dθ

=acoshθand

dx=acoshθdθ