# Find the Integral of Nested Functions

Sometimes you need to integrate a function that is the composition of two functions — for example, the function 2*x* nested inside a sine function. If you were differentiating, you could use the Chain Rule. Unfortunately, no Chain Rule exists for integration.

Fortunately, a function such as

is a good candidate for variable substitution. Follow these steps:

Declare a new variable

*u*as follows and substitute it into the integral:Let

*u*= 2*x*Now substitute

*u*for 2*x*as follows:This may look like the answer to all your troubles, but you have one more problem to resolve. As it stands, the symbol

*dx*tells you that the variable of integration is still*x**.*To integrate properly, you need to find a way to change

*dx*to an expression containing*du**.*That’s what Steps 2 and 3 are about.Differentiate the function

*u*= 2*x*.Substitute 1/2

*du*for*dx*into the integral:You can treat the 1/2 just like any coefficient and use the Constant Multiple Rule to bring it outside the integral:

At this point, you have an expression that you know how to evaluate:

Now that the integration is done, the last step is to substitute 2

*x*back in for*u:*

You can check this solution by differentiating using the Chain Rule: