Chapter 8 - Integration Techniques, L'Hopital's Rule, and Improper Integrals - 8.2 Exercises - Page 522: 67

$$\int x^{n} \sin x d x=-x^{n} \cos x+n \int x^{n-1} \cos x d x$$

Given $$\int x^{n} \sin x d x $$ Use integration by parts , let \begin{aligned} u&= x^n \ \ \ \ \ \ &dv&= \sin x dx\\ du&=nx^{n-1} dx \ \ \ \ \ \ &v&= -\cos x \end{aligned} then \begin{aligned} \int x^{n} \sin x d x &= uv-\int vdu \\ &= -x^n\cos x+n \int x^{n-1}\cos xdx \end{aligned}