Question

Evaluate the integral \[ \int \cos^3 x \cdot \sin x \, dx. \]

Hint:

Use substitution when the integrand contains a function and its derivative.

Answer 1

Use substitution. Let \[ t = \cos x \implies dt = -\sin x \, dx \implies -dt = \sin x \, dx. \] Rewrite the integral: \[ \int \cos^3 x \cdot \sin x \, dx = \int t^3 (-dt) = -\int t^3 \, dt = -\frac{t^4}{4} + C. \] Substitute back: \[ \boxed{ -\frac{\cos^4 x}{4} + C. } \]