Integration by Parts
The Essentials
Integration by parts is used to solve integrals that can not be integrated directly or by using simple u-substitution.
To use integration by parts, split the integral into two pieces and choose one of them to by u and the other to be dv:
\[
\int f(x)g(x)dx=\int u dv
\]
\[
u=f(x), v=g(x)dx
\]
Differentiate u and integrate dv to find du and v. Then, the integral can take the form:
\[
\int udv=uv-\int vdu
\]
Note that choosing which part to be u and which to be v is important.
Example
Evaluate the integral:
\[
\int x^2e^xdx
\]
First, we choose u to be \( x^2 \) and dv to be \( e^x \) :
\[
\begin{array}{cc}
u=x^2 & dv=e^xdx \\
du=2xdx & v=e^x
\end{array}
\]
Plugging these into the formula, we get:
\[
uv-\int vdu
\]
\[
x^2e^x-\int 2xe^xdx
\]
We need to use integration by parts again:
\[
\begin{array}{cc}
u=2x & dv=e^xdx \\
du=2dx & v=e^x
\end{array}
\]
\[
x^2e^x-(\int 2xe^xdx)
\]
\[
x^2e^x-(2xe^x-\int 2e^x)
\]
\[
x^2e^x-2xe^x+2e^x+C
\]
\[
e^x(x^2-2x+2)+C
\]
Practice
Evaluate these expressions:
- \( \int x\sin(x)dx\)
- \( \int xe^xdx\)
- \( \int 5x^2\cos(x)\)
Solutions:
- \( -x\cos(x)+\sin(x)+C \)
- \( xe^x-e^x+C\)
- \( 5x^2\sin(x)+10x\cos(x)-10\sin(x)+C\)