What u-Substitution Does
u-substitution is the integration technique that reverses the chain rule. It is especially useful when the integrand contains a function and its derivative, or something very close to that pattern.
With an indefinite integral, you normally substitute for the inside function and then return to at the end. With a definite integral, however, there is an even cleaner approach: change the limits of integration when you change variables.
Suppose we have an integral of the form
If we let , then , and the integral can often be rewritten completely in terms of .
The key idea is simple:
You change both the variable and the limits. Once you do that, there is no need to substitute back to .
Why the Limits Change
Consider a definite integral whose original limits are expressed in terms of . If you make the substitution , the old limits are no longer limits for . They must be converted.
If the original integral runs from to , then calculate the corresponding -values:
So the transformed integral runs from to .
This is the most important difference between u-substitution with an indefinite integral and u-substitution with a definite integral: the bounds travel with the substitution.
The Step-by-Step Method
- Choose a substitution. Usually choose the complicated inside expression as .
- Find . Differentiate your substitution and solve for the differential needed in the integral.
- Change the lower limit. Plug the original lower bound into your expression for .
- Change the upper limit. Plug the original upper bound into your expression for .
- Rewrite the entire integral in terms of .
- Integrate with respect to .
- Evaluate using the new -limits. Do not substitute back to .
Worked Examples
Evaluate .
Evaluate .
Evaluate .
Evaluate .
The Shortcut: Change the Bounds Immediately
For definite integrals, changing the bounds as soon as you make the substitution is usually the cleanest method.
For example, instead of writing
and eventually returning to , you can transform the limits immediately:
This avoids mixing two variables in the same calculation and eliminates an unnecessary final substitution.
What If You Do Not Change the Limits?
You can perform u-substitution without changing the limits. In that case, you must eventually convert back to before evaluating the original bounds.
For example:
With the substitution , you could find an antiderivative in terms of , substitute back to , and then evaluate at the original endpoints.
That method is mathematically valid, but for definite integrals it creates an extra step. Changing the limits keeps the entire calculation in one variable.
Common Mistakes
Once you replace with , the original numerical limits are still expressed in terms of . Convert them before evaluating the transformed integral.
If you already changed the bounds into -values, there is no need to return to . Evaluate directly using the new limits.
If your substitution is , make sure the expression for actually matches the differential portion of the original integral, including any constant factor.
A good substitution should make the integrand simpler. Look for an inside function whose derivative also appears somewhere in the integrand.
Practice Problems
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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