# Recitation 3

Problem 1: Evaluate $\int \sin^2x \cos^3x dx$.

Hint: Let $u=\sin x$.

Problem 2: Evaluate $\int\sin^2(\pi x)\cos^5(\pi x)dx$.

Hint: Let $u=\sin(\pi x)$.

Problem 3: Evaluate $\int_0^{\pi/2}\cos^2\theta d\theta$.

Hint: Use $\cos^2\theta = \frac{1}{2}(1+\cos 2\theta)$.

Problem 4: Evaluate $\int_0^\pi\cos^4(2t)dt$.

Hint: Use $\cos^2(2t) = \frac{1}{2}(1+\cos 4t)$ and $\cos^2 4t = \frac{1}{2}(1+\cos 8t)$.

Problem 5: Evaluate $\int t\sin^2 tdt$.

Hint: Use $\sin^2 t = \frac{1}{2}(1-\cos 2t)$.

Problem 6: Evaluate $\int \cos\theta\cos^5(\sin\theta) d\theta$

Hint: Let $u=\sin\theta$ and $v=\sin u = \sin(\sin\theta)$.

Problem 7: Evaluate $\int\cos^2 x\tan^3 xdx$.

Hint: Use $\tan x = \sin x / \cos x$ and let $u=\cos x$.

Problem 8: Evaluate $\int \frac{\cos x + \sin 2x}{\sin x}$.

Hint: Use $u=\sin x$ and $\sin 2x = 2\sin x \cos x$.

Problem 9: Prove $\int \sin^n xdx = -\frac{1}{n}\cos x\sin^{n-1}x + \frac{n-1}{n}\int \sin^{n-2}x dx$.

Hint: Apply integral by parts on the left hand side and use $\cos^2 x = 1 - \sin^2 x$.