Mathematical Methods for Physics and Engineering : A Comprehensive Guide

2.2 INTEGRATION

Evaluate the integral

I=

∫

1

x^2 +4x+7

dx.

We can write the integral in the form

I=

∫

1

(x+2)^2 +3

dx.

Substitutingy=x+ 2, we finddy=dxand hence

I=

∫

1

y^2 +3

dy,

Hence, by comparison with the table ofstandard integrals (see subsection 2.2.3)

I=

√

3

3

tan−^1

(

y

√

3

)

+c=

√

3

3

tan−^1

(

x+2

√

3

)

+c.

2.2.8 Integration by parts

Integration by parts is the integration analogy of product differentiation. The

principle is to break down a complicated function into two functions, at least one

of which can be integrated by inspection. The method in fact relies on the result

for the differentiation of a product. Recalling from (2.6) that

d

dx

(uv)=u

dv

dx

+

du

dx

v,

whereuandvare functions ofx, we now integrate to find

uv=

∫

u

dv

dx

dx+

∫

du

dx

vdx.

Rearranging into the standard form for integration by parts gives
∫
u

dv

dx

dx=uv−

∫

du

dx

vdx. (2.36)

Integration by parts is often remembered for practical purposes in the form

the integral of a product of two functions is equal to{the first times the integral of

the second}minus the integral of{the derivative of the first times the integral of

the second}. Here,uis ‘the first’ anddv/dxis ‘the second’; clearly the integralv

of ‘the second’ must be determinable by inspection.

Evaluate the integralI=

∫

xsinxdx.

In the notation given above, we identifyxwithuand sinxwithdv/dx. Hencev=−cosx
anddu/dx= 1 and so using (2.36)

I=x(−cosx)−

∫

(1)(−cosx)dx=−xcosx+sinx+c.