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}.