# Use a Shortcut for Integrating Compositions of Functions

You can use a shortcut to integrate compositions of functions — that is, nested functions of the form *f*(*g*(*x*)). Technically, you’re using the variable substitution *u* = *g*(*x*), but you can bypass this step and still get the right answer.

This shortcut works for compositions of functions *f*(*g*(*x*)) for which

You know how to integrate the outer function

*f**.*The inner function

*g*(*x*)*ax*or*ax*+*b*— that is, it differentiates to a constant.

When these two conditions hold, you can integrate *f*(*g*(*x*)) by using the following three steps:

Write down the reciprocal of the coefficient of

*x.*Multiply by the integral of the outer function, copying the inner function as you would when using the Chain Rule in differentiation.

Add

*C.*

Here’s an example:

Notice that this is a function nested within a function, where the following are true:

The outer function

*f*is the cosine function.The inner function is

*g*(*x*) = 4*x**,*which is of the form*ax**.*

So you can integrate this function quickly as follows:

Write down the reciprocal of 4:

Multiply this reciprocal by the integral of the outer function, copying the inner function:

Add

*C**:*

That’s it! You can check this easily by differentiating, using the Chain Rule:

Here’s another example:

Remember as you begin that sec^{2} 10*x* *dx* is a notational shorthand for [sec (10*x*)]^{2}. So the outer function *f* is the sec^{2} function and the inner function is *g*(*x*) = 10*x.* The criteria for variable substitution are met, so make your way through the steps:

Write down the reciprocal of 10:

Multiply this reciprocal by the integral of the outer function, copying the inner function:

Add

*C:*

Here’s the check:

Take a look at another example:

In this case, the outer function is division, which counts as a function (technically speaking *f*(*x*) = *x*^{–}^{1}). The inner function is 7*x* + 2. Both of these functions meet the criteria, so here’s how to perform this integration:

Write down the reciprocal of the coefficient 7:

Multiply this reciprocal by the integral of the outer function, copying the inner function:

Add

*C**:*

You’re done! You can check your result by differentiating, using the Chain Rule:

Consider one more example:

This time, the outer function *f* is a square root — that is, an exponent of 1/2 — and *g*(*x*) = 12*x* – 5, so you can use a quick substitution:

Write down the reciprocal of 12:

Multiply the integral of the outer function, copying down the inner function:

Add

*C:*

The table gives you a variety of integrals of the form *f*(*g*(*x*)). As you look over this chart, get a sense of the pattern so that you can spot it when you have an opportunity to integrate quickly.