**Exercise:** Use integration by parts to prove the reduction formula \int \tan^nxdx=\frac{\tan^{n-1}x}{n-1}-\int\tan^{n-2}xdx (n\neq 1).

**Exercise:** Evaluate:

- \int_0^1\frac{d}{dx}(e^{\arctan x})dx
- \frac{d}{dx}\int_0^1 e^{\arctan x}dx
- \frac{d}{dx}\int_0^x e^{\arctan t}dt

**Exercise:** Evaluate the integral, if it exists.

- \int\frac{x}{\sqrt{1-x^4}}dx
- \int\frac{\sec\theta\tan\theta}{1+\sec\theta}d\theta
- \int_0^{\pi/4}(1+\tan t)^3\sec^2 tdt

**Exercise:** Find the derivative of the function y=\int_{\sqrt{x}}^x\frac{e^t}{t}dt.