Review 4: Introduction

The goal of this module is to review the integration by parts technique, sometimes shortened as IBP.

In this module you will complete the following activities:

Review the integration by parts technique.
Practice the covered concepts through some exercises, called "Now You Try".

The basic integrals we assume that you know are:

$LaTeX: \begin{align*} &\int u^n\; du =\frac{1}{n+1}u^{n+1}+C,\;\; n \neq -1&\\ &\int u^{-1}\; du= \ln (|u|) +C\\ &\int e^u\; du =e^u+C\\ &\int a^u\; du=\frac{a^u}{\ln a}+C; \hspace{2cm} \mbox{(don't memorize: use}\; \; a^u=e^{u\ln a})\\ &\int \sin(u)\; du= -\cos(u) +C\\ &\int \cos(u)\; du=\sin(u)+C\\ &\int \tan(u)\; du=-\ln|\cos(u)|+C\\ &\int \sec^2(u)=\tan(u)+C\\ &\int \frac{1}{1+u^2} \; du=\arctan(u)+C\\ &\int \frac{1}{\sqrt{1-u^2}} \; du=\arcsin(u)+C \end{align*}$ $\begin{align*} &\int u^n\; du =\frac{1}{n+1}u^{n+1}+C,\;\; n \neq -1&\\ &\int u^{-1}\; du= \ln (|u|) +C\\ &\int e^u\; du =e^u+C\\ &\int a^u\; du=\frac{a^u}{\ln a}+C; \hspace{2cm} \mbox{(don't memorize: use}\; \; a^u=e^{u\ln a})\\ &\int \sin(u)\; du= -\cos(u) +C\\ &\int \cos(u)\; du=\sin(u)+C\\ &\int \tan(u)\; du=-\ln|\cos(u)|+C\\ &\int \sec^2(u)=\tan(u)+C\\ &\int \frac{1}{1+u^2} \; du=\arctan(u)+C\\ &\int \frac{1}{\sqrt{1-u^2}} \; du=\arcsin(u)+C \end{align*}$