Integration by Parts

Integration by Parts

Recall: The product rule.

\dfrac{d}{dx}(uv)=u\dfrac{dv}{dx}+v\dfrac{du}{dx}

Integration by parts is the reverse of this.

{\LARGE \int}u\dfrac{dv}{dx}=uv-{\LARGE \int}v\dfrac{du}{dx}

You should choose u to be something that differentiates nicely, so that you can evaluate the integral on the right-hand side.

Example: Find \int x\sin(x)dx

u=x\;\;\;\dfrac{dv}{dx}=\sin(x)

\dfrac{du}{dx}=1\;\;\;v=-\cos(x)

\begin{aligned}\int x\sin(x)dx&=-x\cos(x)-\int-\cos(x)dx\\[1.2em]&=-x\cos(x)+\int \cos(x)dx\\[1.2em]&=-x\cos(x)+\sin(x)+c\end{aligned}

Integral of the Natural Logarithm

One of the most important uses of integration by parts is that it gives us a way to find \int \ln(x)dx, by writng \ln(x) as
1\times \ln(x).

u=\ln(x)\;\;\;\dfrac{dv}{dx}=1

\dfrac{du}{dx}=\dfrac{1}{x}\;\;\;v=x

\begin{aligned}\int \ln(x)dx&=x\ln(x)-\int x\dfrac{1}{x}dx\\[1.2em]&=x\ln(x)-\int1dx\\[1.2em]&=x\ln(x)-x+c\\[1.2em]&=x(\ln(x)-1)+c\end{aligned}

Integrating by Parts Multiple Times

Some integrals require you to integrate by parts multiple times to get a solution to.

Example: Find \int x^{2}e^{x}dx

u=x^{2}\;\;\;\dfrac{dv}{dx}=e^{x}

\dfrac{du}{dx}=2x\;\;\;v=e^{x}

\begin{aligned}\int x^{2}e^{x}dx=x^{2}e^{x}-\int2xe^{x}dx\end{aligned}

We must integrate by parts again the second integral.

u=2x\;\;\;\dfrac{dv}{dx}=e^{x}

\dfrac{du}{dx}=2\;\;\;v=e^{x}

\begin{aligned}\int x^{2}e^{x}dx&=x^{2}e^{x}-2xe^{x}+\int2e^{x}dx\\[1.2em]&=x^{2}e^{x}-2xe^{x}+2e^{x}+c\end{aligned}