Higher Engineering Mathematics

372 INTEGRAL CALCULUS

=

⎡

⎣θ

3

2

3

2

+

2 θ

1

2

1

2

⎤

⎦

4

1

=

[

2

3

√

θ^3 + 4

√

θ

] 4

1

=

{

2

3

√

(4)^3 + 4

√

4

}

−

{

2

3

√

(1)^3 + 4

√

(1)

}

=

{

16

3

+ 8

}

−

{

2

3

+ 4

}

= 5

1

3

+ 8 −

2

3

− 4 = 8

2

3

Problem 14. Evaluate

∫π

2

0

3 sin 2xdx.

∫π

2

0

3 sin 2xdx

=

[

(3)

(

−

1

2

)

cos 2x

]π

2

0

=

[

−

3

2

cos 2x

]π

2

0

=

{

−

3

2

cos 2

(π

2

)}

−

{

−

3

2

cos 2(0)

}

=

{

−

3

2

cosπ

}

−

{

−

3

2

cos 0

}

=

{

−

3

2

(−1)

}

−

{

−

3

2

(1)

}

=

3

2

+

3

2

= 3

Problem 15. Evaluate

∫ 2

1

4 cos 3tdt.

∫ 2

1

4 cos 3tdt=

[

(4)

(

1

3

)

sin 3t

] 2

1

=

[

4

3

sin 3t

] 2

1

=

{

4

3

sin 6

}

−

{

4

3

sin 3

}

Note that limits of trigonometric functions are always
expressed in radians—thus, for example, sin 6 means
the sine of 6 radians=− 0. 279415 ...

Hence

∫ 2

1

4 cos 3tdt

=

{

4

3

(− 0. 279415 ...)

}

−

{

4

3

(0. 141120 ...)

}

=(− 0 .37255)−(0.18816)=− 0. 5607

Problem 16. Evaluate

(a)

∫ 2

1

4e^2 xdx (b)

∫ 4

1

3

4 u

du,

each correct to 4 significant figures.

(a)

∫ 2

1

4e^2 xdx=

[

4

2

e^2 x

] 2

1

=2[ e^2 x]^21 =2[ e^4 −e^2 ]

=2[54. 5982 − 7 .3891]= 94. 42

(b)

∫ 4

1

3

4 u

du=

[

3

4

lnu

] 4

1

=

3

4

[ln4−ln 1]

=

3

4

[1. 3863 −0]= 1. 040

Now try the following exercise.

Exercise 147 Further problems on definite

integrals

In problems 1 to 8, evaluate the definite inte-

grals (where necessary, correct to 4 significant

figures).

1. (a)

∫ 4

1

5 x^2 dx (b)

∫ 1

− 1

−

3

4

t^2 dt

[

(a) 105 (b)−

1

2

]

2. (a)

∫ 2

− 1

(3−x^2 )dx (b)

∫ 3

1

(x^2 − 4 x+3) dx

[

(a) 6 (b)− 1

1

3

]

3. (a)

∫π

0

3

2

cosθdθ (b)

∫ π

2

0

4 cosθdθ

[(a) 0 (b) 4]

4. (a)

∫π

3

π

6

2 sin 2θdθ (b)

∫ 2

0

3 sintdt

[(a) 1 (b) 4.248]

5. (a)

∫ 1

0

5 cos 3xdx (b)

∫π

6

0

3 sec^22 xdx

[(a) 0.2352 (b) 2.598]