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Chapter 8: Q11-14P (page 459)

Integrate by parts as we did for (11.14) to obtain (11.15) and (11.16).

Short Answer

Expert verified

The integrate by part is 鈭�蠒(x)未n(x-a)dx=(-1)n蠒(a)(n).

Step by step solution

01

Given information.

The given expression is

鈭�蠒(x)未n(x-a)dx

02

Definition of Integration By Parts

Integration by partsor partial integration is a process that finds theintegralof aproductoffunctionsin terms of the integral of the product of theirderivativeandantiderivative.

03

Determine the value of the given expression

The term which is to prove is

鈭�蠒(x)未x(x-a)dx=蠒n(a)

Consider 鈭�蠒(x)未n(x-a)dxintegrate by parts

Thus, again the integration by parts multiplied a negative sign, and differentiated the test function once and integrated the delta function one time, thus we expect that using the integration by parts times, we get

I=(-1)n鈭�-鈭�鈭�蠒(x)n未(x-a)(n-n)dx

Where,

未(x-a)(0)=未(x-a)

Thus, we have

I=(-1)n鈭�-鈭�鈭�蠒(x)(n)未(x-a)dx

And, we evaluate the integral, thus we have

I=(-1)n蠒(a)(n)

Thus, the integrate by part is 鈭�蠒(x)未n(x-a)dx=(-1)n蠒(a)(n)


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