Chapter 9, Problem 1CCE

Describe integration by substitution in your own words.

To determine

Integration by substitution.

Answer to Problem 1CCE

Solution:

The reverse process of the chain rule of derivative gives integration by substitution.

Explanation of Solution

Given information:

Instructions to describe integration by substitution

Explanation:

If $f(x)\text{and}g(x)$ be two given functions and $F(x)$ be an anti-derivative for $f(x)$. The chain rule gives:

$\frac{d}{dx}\left[F\left(g(x)\right)\right]=F\text{'}\left(g(x)\right)g\text{'}(x)\Rightarrow f\left(g(x)\right)g\text{'}(x)$.

Turning this formula into an integration formula, it will be

${\displaystyle \int f\left(g(x)\right)g\text{'}dx=F\left(g(x)\right)+C}$, where $C$ is any constant

Therefore, the reverse process of the chain rule gives integration by substitution.