# How to Find Antiderivatives with the Substitution Method

When a function’s argument (that’s the function’s input) is more complicated than something like 3*x* + 2 (a *linear* function of *x** *— that is, a function where *x* is raised to the first power), you can use the substitution method. This method works when the integrand contains a function* and the derivative of the function’s argument* — in other words, when it contains that extra thing produced by the chain rule — or something just like it except for a constant. And the integrand must *not* contain anything else. (If that sounds like gibberish, it’ll become clear when you read the following example).

Find the derivative of

with the substitution method.

Set

*u*equal to the argument of the main function.Take the derivative of

*u*with respect to*x.*Solve for

*dx.*Make the substitutions.

Antidifferentiate by using the simple reverse rule.

Substitute

*x*-squared back in for*u*— coming full circle.

If the original problem had been

Now, you finish this problem just as you did in the preceding Steps 5 and 6, except for the extra 5/2.

Because *C* is any old constant,

You should check this by differentiating it.