Integrating Using Integration by Parts

#6.11 Integrating Using Integration by Parts

Welcome back to AP Calculus with Fiveable! In this study guide, we'll delve into the technique of Integration by Parts. Integration by Parts is a powerful method used to integrate the product of two functions, and it often comes in handy when dealing with more complex integrals. We have a few techniques such as u-substitution and Riemann sums in our calculus toolbox, so let's keep building those integration skills! 🧱

#🔄 Integration by Parts Basics

Take a look at the following integral:

$$\int x^2 \sin(x),dx$$

We can’t use any of our current integration tools to evaluate it: substitution fails, and we only know how to integrate $x^2$ and $\sin(x)$ separately. Why don’t we try to use that to our advantage?

For the integral above, we will be using a method called Integration by Parts, which is based on the product rule for differentiation. It’s essentially the reverse process!

Here is the product rule, as $u$ and $v$ representing two different functions.

$$\frac{d}{dx} uv = \textcolor{blue}{u}\cdot \textcolor{pink}{v'} +\textcolor{red}{ v}\cdot \textcolor{teal}{u'}$$

If we try to reverse the process, we take the integral of all of those terms. It will look like the following:

$$uv = \int\textcolor{blue}{u}\cdot \textcolor{pink}{v'} +\int\textcolor{red}{ v}\cdot \textcolor{teal}{u'}$$

When we rearrange the terms, we get the following rule for Integration by Parts:

$$\int \textcolor{blue}{u} , \textcolor{pink}{dv} = uv - \int \textcolor{red}{v} , \textcolor{teal}{du}$$

Where:

• $\textcolor{blue}{u}$ and $\textcolor{pink}{dv}$ are selected parts of the integrand.
• $\textcolor{teal}{du}$ is the derivative of $\textcolor{blue}{u}$...