Higher Engineering Mathematics

370 INTEGRAL CALCULUS

(a) From Table 37.1(iv),
∫
7 sec^24 tdt=(7)

(

1

4

)

tan 4t+c

=

7

4

tan 4t+c

(b) From Table 37.1(v),

3

∫

cosec^22 θdθ=(3)

(

−

1

2

)

cot 2θ+c

=−

3

2

cot 2θ+c

Problem 10. Determine

(a)

∫

5e^3 xdx (b)

∫

2

3e^4 t

dt.

(a) From Table 37.1(viii),
∫
5e^3 xdx=(5)

(

1

3

)

e^3 x+c=

5

3

e^3 x+c

(b)

∫

2

3e^4 t

dt=

∫

2

3

e−^4 tdt=

(

2

3

)(

−

1

4

)

e−^4 t+c

=−

1

6

e−^4 t+c=−

1

6e^4 t

+c

Problem 11. Determine

(a)

∫

3

5 x

dx (b)

∫(

2 m^2 + 1

m

)

dm.

(a)

∫

3

5 x

dx=

∫(

3

5

)(

1

x

)

dx=

3

5

lnx+c

(from Table 37.1(ix))

(b)

∫(

2 m^2 + 1

m

)

dm=

∫(

2 m^2

m

+

1

m

)

dm

=

∫(

2 m+

1

m

)

dm

=

2 m^2

2

+lnm+c

=m^2 +lnm+c

Now try the following exercise.

Exercise 146 Further problems on standard

integrals

In Problems 1 to 12, determine the indefinite

integrals.

1. (a)

∫

4dx (b)

∫

7 xdx

[

(a) 4x+c (b)

7 x^2

2

+c

]

2. (a)

∫

2

5

x^2 dx (b)

∫

5

6

x^3 dx

[

(a)

2

15

x^3 +c (b)

5

24

x^4 +c

]

3. (a)

∫(

3 x^2 − 5 x

x

)

dx (b)

∫

(2+θ)^2 dθ

⎡

⎢

⎢

⎣

(a)

3 x^2

2

− 5 x+c

(b) 4θ+ 2 θ^2 +

θ^3

3

+c

⎤

⎥

⎥

⎦

4. (a)

∫

4

3 x^2

dx (b)

∫

3

4 x^4

dx

[

(a)

− 4

3 x

+c (b)

− 1

4 x^3

+c

]

5. (a) 2

∫√

x^3 dx (b)

∫

1

4

√ 4

x^5 dx

[

(a)

4

5

√

x^5 +c (b)

1

9

√ 4

x^9 +c

]

6. (a)

∫

− 5

√

t^3

dt (b)

∫

3

7

√ 5

x^4

dx

[

(a)

10

√

t

+c (b)

15

7

√ (^5) x+c
]

7. (a)

∫

3 cos 2xdx (b)

∫

7 sin 3θdθ

⎡

⎢

⎢

⎣

(a)

3

2

sin 2x+c

(b)−

7

3

cos 3θ+c

⎤

⎥

⎥

⎦