Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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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.
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