# 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 $y=\int_{\sqrt{x}}^x\frac{e^t}{t}dt$.