## Review: Integration

To ensure success in this course, your integration skills must be flawless. Practice with the following exercises, and discuss among yourselves the best techniques to evaluate the integrals below. Feel free to drop questions and comments, and I will try to guide you in the right direction—without offering the solution, of course; that will spoil the fun for others that want to try by themselves.

 $\displaystyle{\int_0^3 \frac{dx}{x-1} }$ $\displaystyle{\int_{-\infty}^0 xe^{-x}\, dx }$ $\displaystyle{\int \cos x \big( 1+\sin^2 x \big)\, dx }$ $\displaystyle{\int \frac{\sin x + \sec x}{\tan x}\, dx }$ $\displaystyle{\int_1^3 r^4 \ln r\, dr }$ $\displaystyle{\int \frac{x-1}{x^2-4x+5}\, dx }$ $\displaystyle{\int \sin^3 \theta \cos^5 \theta\, d\theta }$ $\displaystyle{\int x\, \sin^2 x\, dx }$ $\displaystyle{\int e^{x+e^x}\, dx }$ $\displaystyle{\int e^2\, dx }$ $\displaystyle{\int \frac{\ln x}{x \sqrt{1+ \big(\ln x \big)^2}}\, dx }$ $\displaystyle{\int \big( 1+ \sqrt{x} \big)^8\, dx }$ $\displaystyle{\int \ln \big( x^2-1 \big)\, dx }$ $\displaystyle{\int \frac{3x^2-2}{x^2-2x-8}\, dx }$ $\displaystyle{\int \frac{dx}{1+e^x} }$ $\displaystyle{\int \sqrt{3-2x-x^2}\, dx }$ $\displaystyle{\int \frac{1+\cot x}{4-\cot x}\, dx }$ $\displaystyle{\int \sin 4x \cos 3x\, dx }$ $\displaystyle{\int e^x \sqrt{1+e^x}\, dx }$ $\displaystyle{\int \sqrt{1+e^x}\, dx }$ $\displaystyle{\int x^5 e^{-x^3}\, dx }$ $\displaystyle{\int \frac{1+\sin x}{1-\sin x}\, dx }$ $\displaystyle{\int \frac{dx}{3-5\sin x} }$ $\displaystyle{\int \frac{dx}{3\sin x - 4\cos x} }$
1. October 31, 2011 at 10:31 am

For number two I changed the negative infinity to the variable t and then took the limit as t approaches negative infinity of the integral. I got an answer of infinity, but wolfram alpha states that the “integral does not converge”. Did I get the right answer then? When I calculate the limit as infinity does that mean that the integral does not converge?

• October 31, 2011 at 11:11 am

They are the same thing. See section 7.8 on improper integrals.

2. January 26, 2012 at 2:18 pm

Are there supposed to be two dx’s in number one?

• January 28, 2012 at 8:39 am

Nope, good catch!

• October 15, 2012 at 8:05 pm