Chapter 8, Problem 1GYR

To determine

What is the formula for integration by parts and its origin and suggest the reasons behind its usage.

Explanation of Solution

Write the formula for integration by parts as below.

${\displaystyle \int u\left(x\right)v^{\prime }\left(x\right)dx}=u\left(x\right)v\left(x\right)-{\displaystyle \int v\left(x\right)u^{\prime }\left(x\right)dx}$

The integration by parts formula has been derived from the product rule in the integral form as below.

$\frac{d}{dx}\left[u\left(x\right)v\left(x\right)\right]=u^{\prime }\left(x\right)v\left(x\right)+u\left(x\right)v^{\prime }\left(x\right)$

Integrate the equation as below.

${\displaystyle \int \frac{d}{dx}\left[u\left(x\right)v\left(x\right)\right]dx}={\displaystyle \int u^{\prime }\left(x\right)v\left(x\right)dx}+{\displaystyle \int u\left(x\right)v^{\prime }\left(x\right)dx}$

Rearrange the terms as below.

${\displaystyle \int u\left(x\right)v^{\prime }\left(x\right)dx}={\displaystyle \int \frac{d}{dx}\left[u\left(x\right)v\left(x\right)\right]dx}-{\displaystyle \int v\left(x\right)u^{\prime }\left(x\right)dx}$

Simplify it as below.

${\displaystyle \int u\left(x\right)v^{\prime }\left(x\right)dx}=\left[u\left(x\right)v\left(x\right)\right]-{\displaystyle \int v\left(x\right)u^{\prime }\left(x\right)dx}$

Thus, the formula for integration by parts comes from the product rule in integral form.

The formula is helpful to solve the integral present in the form ${\displaystyle \int u\left(x\right)v^{\prime }\left(x\right)dx}$ and is also difficult to solve directly. Hence, the solution will be obtained by solving the integral term ${\displaystyle \int v\left(x\right)u^{\prime }\left(x\right)dx}$, which is much easier to be computed.

This formula is useful when the function u can be differentiated repeatedly and term $v^{\prime }$ can be integrated repeatedly without any difficulty.

Therefore, the formula for integration by parts is ${\displaystyle \int u\left(x\right)v^{\prime }\left(x\right)dx}=\left[u\left(x\right)v\left(x\right)\right]-{\displaystyle \int v\left(x\right)u^{\prime }\left(x\right)dx}$. It comes from writing the product rule in integral form. It is useful in conditions where the integral form ${\displaystyle \int u\left(x\right)v^{\prime }\left(x\right)dx}$ is difficult to solve compared to solving the term ${\displaystyle \int v\left(x\right)u^{\prime }\left(x\right)dx}$.