Integration by Parts Review

The Rule: $\int u\mathrm{d}v=uv-\int v\mathrm{d}u$

Choose the best choice for $u$ by picking a function that becomes easier to integrate after differentiating Let $v$ be everything in the integrand other than $u$

Example 1: Find $\int x{e}^x\mathrm{d}x$

Step 1: Choose $u$ and $v$

Since $e^x$ remains the same after differentiation, let’s choose $x$ as our $u$ Let’s let $v$ equal everything in the integrand other than $x$ So, after this, we should be left with:

$$\begin{array}{r}\mathrm{let}u=x\mathrm{let}\mathrm{d}v=e^x\mathrm{d}x\\ \mathrm{d}u=\mathrm{d}xv=e^x\mathrm{d}x\end{array}$$

Step 2: Solve using parts

$$\begin{array}{r}\int x{e}^x\mathrm{d}x=x{e}^x-\int e^x\mathrm{d}x\\ =x{e}^x-e^x+C\end{array}$$

The Derivation:

Note: Integration by parts