Integration By Substitution

Reference > Calculus: Integration

Description

A method of integration that simplifies the function by rewriting it in terms of a different variable. The goal is to transform the integral into another that is easier to solve. It is the inverse of the chain rule in differentiation.

The new variable used is usually \(u\) by convention, hence this method is also known as 'U-Substitution'.


Examples
\[\int (\sin{({x}^{2})})x \, dx\]
1
Regroup terms
\[\int x\sin{({x}^{2})} \, dx\]

2
Let \(u={x}^{2}\), \(du=2x dx\), then \(x dx=\frac{1}{2} du\)

3
Using \(u\) and \(du\) above, rewrite \(\int x\sin{({x}^{2})} \, dx\)
\[\int \frac{\sin{u}}{2} \, du\]

4
Use Constant Factor Rule: \(\int cf(x) \, dx=c\int f(x) \, dx\)
\[\frac{1}{2}\int \sin{u} \, du\]

5
The integral of \(\sin{u}\) is \(-\cos{u}\)
\[-\frac{\cos{u}}{2}\]

6
Substitute \(u={x}^{2}\) back into the original integral
\[-\frac{\cos{({x}^{2})}}{2}\]

7
Add constant
\[-\frac{\cos{({x}^{2})}}{2}+C\]

Done