Recitation 24

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:

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

Exercise: Evaluate the integral, if it exists.

1. \int\frac{x}{\sqrt{1-x^4}}dx
2. \int\frac{\sec\theta\tan\theta}{1+\sec\theta}d\theta
3. \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.