**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.