Integration by substitution

In calculus, integration by substitution — popularly called u-substituion or simply the substitution method — is a technique of integration whereby a complicated looking integrand is rewritten into a simpler form by using a change of variables:

$\int f\bigl(g(x)\bigr)g'(x)dx=\int f(u)du$ , where $u=g(x)$ .

In the case of a definite integral,

$\int\limits_a^b f\bigl(g(x)\bigr)g'(x)dx=\int\limits_c^d f(u)du$ , where $u=g(x)$ , $c=g(a),\ d=g(b)$ .

Integration by u-substitution is the inverse operation of the chain rule from differential calculus. It is also the two-dimensional version of using a Jacobian matrix to transform coordinates.

Example

Consider the integral:

$\int x(x+3)^7dx$

By letting $u=x+3$ , thus $du=dx$ (since $\frac{du}{dx}=1$), and observing that $x=u-3$ , the integral simplifies to

$\int x(x+3)^7dx=\int(u-3)u^7du=\int(u^8-3u^7)du$

which is easily integrated to obtain:

$\frac{u^9}{9}-\frac{3u^8}{8}+C=\frac{(x+3)^9}{9}-\frac{3(x+3)^8}{8}+C$

Note that this integral can also be done using integration by parts, although the final answer may look different because of the different steps involved.