When to Use Variable Substitution with Integrals
Variable substitution comes in handy for some integrals. The antidifferentiation formulas plus the Sum Rule, Constant Multiple Rule, and Power Rule allow you to integrate a variety of common functions. But as functions begin to get a little bit more complex, these methods become insufficient. For example, these methods don’t work on the following:
To evaluate this integral, you need some stronger medicine. The sticking point here is the presence of the constant 2 inside the sine function. You have an antidifferentiation rule for integrating the sine of a variable, but how do you integrate the sine of a variable times a constant?
The answer is variable substitution, a fivestep process that allows you to integrate where no integral has gone before. Here are the steps:

Declare a variable u and set it equal to an algebraic expression that appears in the integral, and then substitute u for this expression in the integral.

Differentiate u to find du/dx.
This gives you the differential du = ƒ'(x)dx.

Make another substitution to change dx and all other occurrences of x in the integral to an expression that includes du.

Integrate using u as your new variable of integration.

Express this answer in terms of x.