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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:

$ \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.

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