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When to Use Variable Substitution with Integrals

Variable substitution comes in handy for some integrals. The anti-differentiation 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:

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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 anti-differentiation 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 five-step process that allows you to integrate where no integral has gone before. Here are the steps:

  1. 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.

  2. Differentiate u to find du/dx.

    This gives you the differential du = ƒ'(x)dx.

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

  4. Integrate using u as your new variable of integration.

  5. Express this answer in terms of x.

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