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.

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