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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 ∫ϕ(x)δn(x-a)dx=(-1)nϕ(a)(n).

Step by step solution

01

Given information.

The given expression is

∫ϕ(x)δn(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

∫ϕ(x)δx(x-a)dx=ϕn(a)

Consider ∫ϕ(x)δn(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∫-∞∞ϕ(x)nδ(x-a)(n-n)dx

Where,

δ(x-a)(0)=δ(x-a)

Thus, we have

I=(-1)n∫-∞∞ϕ(x)(n)δ(x-a)dx

And, we evaluate the integral, thus we have

I=(-1)nϕ(a)(n)

Thus, the integrate by part is ∫ϕ(x)δn(x-a)dx=(-1)nϕ(a)(n)


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