Integration By Parts

The technique of integrating by parts borrows from differentiation’s Product rule.

Definition §

Assume we have two differentiable functions, $f$ and $g$. If we let $f=u$ and $g=v$, we can solve the integral of the form:

$$\int f(x)g^{\prime }(x)dx=\int u\frac{dv}{dx}dx=uv-\int \frac{du}{dx}vdx$$

Why?

The Product rule states that: $\frac{d}{dx}(uv)=\frac{du}{dx}v+\frac{dv}{dx}u$ By integrating both sides: $\int \frac{d}{dx}(uv)dx=\int \frac{du}{dx}v+\frac{dv}{dx}udx$ $uv=\int \frac{du}{dx}vdx+\int \frac{dv}{dx}udx$ $\int u\frac{dv}{dx}dx=uv-\int \frac{du}{dx}vdx$

Examples §

1: Exponentials & Polynomials

Solve $\int x{e}^{5x}dx$
Let $u=x$ and $\frac{dv}{dx}=e^{5x}$. Then we can represent the integral as: $\int u\frac{dv}{dx}dx$ $\frac{du}{dx}=1$ and $v=\frac{1}{5}{e}^{5x}$

Using integration by parts:

$\int u\frac{dv}{dx}dx=uv-\int \frac{du}{dx}vdx$ $=x\times \frac{1}{5}{e}^{5x}-\int 1\times \frac{1}{5}{e}^{5x}dx$

$=\frac{1}{5}x{e}^{5x}-\frac{1}{25}{e}^{5x}+c$