Basic Engineering Mathematics, Fifth Edition

326 Basic Engineering Mathematics

(b) When a sum of several terms is integrated the

result is the sum of the integrals of the separate

terms. For example,

∫

( 3 x+ 2 x^2 − 5 )dx

=

∫

3 xdx+

∫

2 x^2 dx−

∫

5 dx

=

3 x^2

2

+

2 x^3

3

− 5 x+c

35.3 Standard integrals

From Chapter 34,

d

dx

(sinax)=acosax. Since inte-

gration is the reverse process of differentiation, it

follows that

∫

acosax dx=sinax+c

or

∫

cosax dx=

1

a

sinax+c

By similar reasoning

∫

sinax dx=−

1

a

cosax+c

∫

eaxdx=

1

a

eax+c

and

∫

1

x

dx=lnx+c

From above,

∫

axndx=

axn+^1

n+ 1

+cexcept when

n=− 1

Whenn=− 1 ,

∫

x−^1 dx=

∫ 1

x

dx=lnx+c

A list of standard integrals is summarized in

Table 35.1.

Table 35.1Standard integrals

y

∫

ydx

1.

∫

axn

axn+^1

n+ 1

+c(except whenn=− 1 )

2.

∫

cosax dx

1

a

sinax+c

3.

∫

sinax dx −

1

a

cosax+c

4.

∫

eaxdx

1

a

eax+c

5.

∫ 1

x

dx lnx+c

Problem 1. Determine

∫

7 x^2 dx

The standard integral,

∫

axndx=

axn+^1

n+ 1

+c

Whena=7andn= 2 ,

∫

7 x^2 dx=

7 x^2 +^1

2 + 1

+c=

7 x^3

3

+c or

7

3

x^3 +c

Problem 2. Determine

∫

2 t^3 dt

Whena=2andn= 3 ,

∫

2 t^3 dt=

2 t^3 +^1

3 + 1

+c=

2 t^4

4

+c=

1

2

t^4 +c

Note that each of the results in worked examples 1 and

2 may be checked by differentiating them.

Problem 3. Determine

∫

8 dx

∫

8 dxis the same as

∫

8 x^0 dxand, using the general

rule whena=8andn=0, gives

∫

8 x^0 dx=

8 x^0 +^1

0 + 1

+c= 8 x+c

In general, ifkis a constant then

∫

kdx=kx+c.

Problem 4. Determine

∫

2 xdx

Whena=2andn= 1 ,

∫

2 xdx=

∫

2 x^1 dx=

2 x^1 +^1

1 + 1

+c=

2 x^2

2

+c

=x^2 +c

Problem 5. Determine

∫ (

3 +

2

5

x− 6 x^2

)

dx

∫

(

3 +

2

5

x− 6 x^2

)

dxmay be written as

∫

3 dx+

∫ 2

5

xdx−

∫

6 x^2 dx

i.e., each term is integrated separately. (This splitting

up of terms only applies, however, for addition and