## FANDOM

1,168 Pages

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.

Community content is available under CC-BY-SA unless otherwise noted.