Basic Engineering Mathematics, Fifth Edition

328 Basic Engineering Mathematics

From 3 of Table 35.1,

∫

5sin2θdθ=( 5 )

(

−

1

2

)

cos2θ+c

=−

5

2

cos 2θ+c

Problem 13. Determine

∫

5 e^3 xdx

From 4 of Table 35.1,

∫

5 e^3 xdx=( 5 )

(

1

3

)

e^3 x+c

=

5

3

e^3 x+c

Problem 14. Determine

∫

2

3 e^4 t

dt

∫

2

3 e^4 t

dt=

∫

2

3

e−^4 tdt

=

(

2

3

)(

−

1

4

)

e−^4 t+c

=−

1

6

e−^4 t+c=−

1

6 e^4 t

+c

Problem 15. Determine

∫

3

5 x

dx

From 5 of Table 35.1,

∫

3

5 x

dx=

∫ (

3

5

)(

1

x

)

dx

=

3

5

lnx+c

Problem 16. Determine

∫ (

2 x^2 + 1

x

)

dx

∫ (

2 x^2 + 1

x

)

dx=

∫ (

2 x^2

x

+

1

x

)

dx

=

∫ (

2 x+

1

x

)

dx=

2 x^2

2

+lnx+c

=x^2 +lnx+c

Now try the following Practice Exercise

PracticeExercise 139 Standard integrals

(answers on page 355)

Determine the following integrals.

1. (a)

∫

4 dx (b)

∫

7 xdx

2. (a)

∫

5 x^3 dx (b)

∫

3 t^7 dt

3. (a)

∫ 2

5

x^2 dx (b)

∫ 5

6

x^3 dx

4. (a)

∫

( 2 x^4 − 3 x)dx (b)

∫

( 2 − 3 t^3 )dt

5. (a)

∫

(

3 x^2 − 5 x

x

)

dx (b)

∫

( 2 +θ)^2 dθ]

6. (a)

∫

( 2 +θ)( 3 θ− 1 )dθ

(b)

∫

( 3 x− 2 )(x^2 + 1 )dx

7. (a)

∫ 4

3 x^2

dx (b)

∫ 3

4 x^4

dx

8. (a) 2

∫√

x^3 dx (b)

∫ 1

4

√ 4

x^5 dx

9. (a)

∫ − 5

√

t^3

dt (b)

∫ 3

7

√ 5

x^4

dx

10. (a)

∫

3cos2xdx (b)

∫

7sin3θdθ

11. (a)

∫

3sin

1

2

xdx (b)

∫

6cos

1

3

xdx

12. (a)

∫ 3

4

e^2 xdx (b)

2

3

∫ dx

e^5 x

13. (a)

∫ 2

3 x

dx (b)

∫

(

u^2 − 1

u

)

du

14. (a)

∫ ( 2 + 3 x)^2

√

x

dx (b)

∫

(

1

t

+ 2 t

) 2

dt

35.4 Definite integrals

Integrals containing an arbitrary constant cin their

results are calledindefinite integralssince theirprecise

value cannot be determined without furtherinformation.

Definite integralsare those in which limits are applied.

Ifanexpressioniswrittenas[x]ba, biscalledtheupper

limitandathelower limit. The operation of applying

the limits is defined as [x]ba=(b)−(a).