How to Recognize a u-Substitution Problem
u-substitution is essentially the reverse of the chain rule. When you see a complicated expression inside another function, look for the derivative of that inside expression somewhere else in the integrand.
For example, consider . The expression is the natural candidate for because its derivative is , which also appears in the integral.
The goal is to transform the original integral into an integral involving only . Once the substitution is made, the resulting integral should usually be simpler than the original.
The Basic Method
- Choose . Usually, choose the complicated expression that is "inside" another function.
- Find . Differentiate your choice of with respect to .
- Rewrite the integral in terms of . Replace the relevant part of the integrand with .
- Integrate with respect to .
- Substitute back. For an indefinite integral, return to the original variable.
Don't automatically choose the first expression you see. Ask: "What function would have a derivative that also appears in the integral?" The best substitution is the one that makes the entire integral simpler.
Worked Example
Evaluate .
Definite Integrals
u-substitution also works with definite integrals. There are two valid approaches. You can substitute back to before applying the original bounds, or you can change the bounds to match the new variable.
Changing the bounds is often cleaner. If and the original bounds are , then calculate the corresponding -values before integrating.
Practice Problems
Try each problem before opening the solution. The questions are arranged so that the substitution becomes progressively less obvious.
Evaluate .
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Evaluate .
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Evaluate .
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Evaluate .
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Evaluate .
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Evaluate .
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Evaluate .
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Evaluate .
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Evaluate .
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Evaluate .
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Common Mistakes
Not every expression in an integral is a good choice for . Choose an expression whose derivative appears elsewhere in the integrand. If your substitution leaves the integral looking just as complicated as before, reconsider it.
If , then , so . You must account for that constant when converting the original integral.
For an indefinite integral, the final answer should normally be written in terms of , not . Leaving the answer in terms of is incomplete unless the problem specifically allows it.
If you change variables from to , either change the bounds as well or substitute back before evaluating. Do not mix the original -bounds with an integral written entirely in terms of .
How to Check Your Answer
The most powerful check for an indefinite u-substitution problem is to differentiate your answer. If differentiating your result returns the original integrand, your antiderivative is correct.
This is particularly useful on AP Calculus problems because u-substitution is really just the chain rule running backward. Differentiation gives you a direct way to verify the result.
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