Integration by parts

Integration by parts is the product rule $(fg)'=f'g+fg'$ integrated and rearranged.

The key decision is which factor to differentiate: choose the one that gets simpler ($x$ or $\log x$) as $f$. For $x\log x$, differentiate $\log x$.

Examples

1. 01What is f(-1)?$\displaystyle f(x)=x^2$AnswerHide

   Answer$\displaystyle 1$
2. 02What is the $uv$ part?$\displaystyle \int x e^x \, dx \quad (u=x,\; v'=e^{x} \;\Rightarrow\; v=e^{x})$AnswerHide

   Answer$\displaystyle x e^x$
3. 03What is f(0)?$\displaystyle f(x)=3x+3$AnswerHide

   Answer$\displaystyle 3$
4. 04What is the $uv$ part?$\displaystyle \int x \log_e x \, dx \quad (u=\log_e x,\; v'=x \;\Rightarrow\; v=\frac{x^{2}}{2})$AnswerHide

   Answer$\displaystyle \frac{x^{2}}{2} \log_e x$
5. 05What is the $uv$ part?$\displaystyle \int x \sin x \, dx \quad (u=x,\; v'=\sin x \;\Rightarrow\; v=-\cos x)$AnswerHide

   Answer$\displaystyle -x \cos x$
6. 06What is $\int u'v\,dx$? (omit $C$)$\displaystyle \int x \log_e x \, dx = \frac{x^{2}}{2} \log_e x - \int u'v \, dx$AnswerHide

   Answer$\displaystyle \frac{x^{2}}{4}$

FAQ

Q1How do I choose f and g'?Show answerHide

A

Make f the factor that gets simpler when differentiated. $\log x$ and x get simpler, so they go on the f side; $e^x$ and $\sin x$ are easy to integrate, so they go on the $g'$ side.

Practice this pattern in the app, 5 problems a day.

Published 2026-09-03 · Updated 2026-09-05