Chapter 7.3, Problem 48E

To determine

To establish the reduction formula for the integral ${\displaystyle \int x^n\sin xdx}=-x^n\cos x+n{\displaystyle \int x^{n-1}\cos xdx}$

Explanation of Solution

Given information: ${\displaystyle \int x^n\sin xdx}$

Formula used: ${\displaystyle \int uvdx=u{\displaystyle \int vdx-{\displaystyle \int \left(u^{\prime }{\displaystyle \int vdx}\right)}}}dx$

Calculation :

Let $I_n={\displaystyle \int x^n\sin xdx}$

  $={\displaystyle \int x^n\sin xdx}=x^n(-\cos x)-n{\displaystyle \int x^{n-1}(-\cos x)dx}+C$

which establishes the expression