# Recitation 7

Example 1: Find $\int\frac{1}{(1+x^2)^2}dx$.

Hint: Use $x=\tan\theta$.

Example 2: Find $\int\sqrt{\frac{1-x}{1+x}}dx$.

Hint: Observe that $\sqrt{\frac{1-x}{1+x}}=\frac{1-x}{\sqrt{1-x^2}}=\frac{1}{\sqrt{1-x^2}}+\frac{-x}{\sqrt{1-x^2}}$.

Problem 3: Find $\int_0^\pi t\cos^2 tdt$.

Hint: Note that the integral is equal to $\int_0^\pi t\frac{1+\cos 2t}{2}dt = \frac{1}{2}\int_0^\pi t dt + \frac{1}{2}\int_0^\pi t\cos 2t dt$.

Problem 4: Find $\int_0^1 (1+\sqrt{x})^8 dx$.

Hint: Use $u=1+\sqrt{x}$.

Problem 5: Find $\int\theta\tan^2\theta d\theta$.

Hint: Observe the integral is equal to $\int\theta(\sec^2\theta - 1)d\theta = \int\theta d\tan\theta - \theta^2/2$.

Problem 6: Find $\int\frac{1}{x\sqrt{4x+1}}dx$.

Hint: Use $u=\sqrt{4x+1}$. Then $4x = u^2 -1, 4dx = 2udu$ and the integral becomes $\int\frac{2}{u^2-1}du$.

Problem 7: Find $\int x^5 e^{-x^3}dx$.

Hint: Use $u=-x^3$.

Problem 8: Find $\int x^3\sqrt{x+c}dx$.

Hint: Use $u=x+c$.

Problem 9: Find $\int \sqrt{x} e^{\sqrt{x}}dx$.

Hint: Use $u=\sqrt{x}$.