Question

Use the product rule to derive the formula for integration by parts in theorem 5.7.

Short answer

The solution is$\int uvdx=u\int vdx-\int \left[\frac{d}{dx}\left(u\right)\int vdx\right]dx$.

Step-by-step solution

Step 1. Given information.

The product rule of differentiation.

$\frac{d}{dx}\left(uv\right)=u\frac{d}{dx}\left(v\right)+v\frac{d}{dx}\left(u\right)$

Step 2. Derivation of the formula for integration by parts.

If $y=uv$then, $\frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}$.

Rearranging this rule:

$u\frac{dv}{dx}=\frac{d\left(uv\right)}{dx}-v\frac{du}{dx}$

Now integrate both sides:

$\int u\frac{dv}{dx}dx=\int \frac{d\left(uv\right)}{dx}dx-\int v\frac{du}{dx}dx$

The first term on the right simplifies since we are simply integrating.

$\int u\frac{dv}{dx}dx=uv-\int v\frac{du}{dx}dx$

Step 3. Simplified answer.

This is the formula known as integration by parts.

$\int u\frac{dv}{dx}dx=uv-\int v\frac{du}{dx}dx$