Calculus: Early Transcendentals, Chapter 5, 5.5, Section 5.5, Problem 5

You need to evaluate the indefinite integral by performing the substitution u =cos theta , such that:
u = cos theta => du = -sin theta*d theta => sin theta*d theta = -du
int cos^3 theta*sin theta*d theta = - int u^3 du
Using the formula int u^n du = (u^(n+1))/(n+1) + c yields
- int u^3 du = -(u^(3+1))/(3+1) + c
- int u^3 du = -(u^4)/4 + c
Replacing back cos theta for u yields:
int cos^3 theta*sin theta*d theta = -(cos^4 theta)/4 + c
Hence, evaluating the indefinite integral yields int cos^3 theta*sin theta*d theta = -(cos^4 theta)/4 + c.