# Integration by substitution

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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:

- , where .

In the case of a definite integral,

- , where , .

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:

By letting , thus (since ), and observing that , the integral simplifies to

which is easily integrated to obtain:

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.