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

Regroup terms.

\int x\sin{({x}^{2})} \, dx

Use Integration by Substitution.

Let

u={x}^{2}

du=2x \, dx

, then

x \, dx=\frac{1}{2} \, du

Using

u

and

du

above, rewrite

\int x\sin{({x}^{2})} \, dx

\int \frac{\sin{u}}{2} \, du

Use Constant Factor Rule:

\int cf(x) \, dx=c\int f(x) \, dx

\frac{1}{2}\int \sin{u} \, du

Use Trigonometric Integration: the integral of

\sin{u}

-\cos{u}

-\frac{\cos{u}}{2}

Substitute

u={x}^{2}

back into the original integral.

-\frac{\cos{({x}^{2})}}{2}

Add constant.

-\frac{\cos{({x}^{2})}}{2}+C

Done