Download Article
Download Article
X
wikiHow is a “wiki,” similar to Wikipedia, which means that many of our articles are cowritten by multiple authors. To create this article, volunteer authors worked to edit and improve it over time.
This article has been viewed 16,965 times.
Learn more...
When you encounter a function nested within another function, you cannot integrate as you normally would. In that case, you must use usubstitution.
Steps
Part 1
Part 1 of 3:Indefinite Integral
Download Article
Part 1

1Determine what you will use as u. Finding u may be the most difficult part of usubstitution, but as you practice, it will become more natural. In general, a good usub involves the derivative of u cancelling out part of the integrand. The easiest integrals are those where it includes a function (any multiple of ) nested within another elementary function  in these cases, the nested function will be u.
 Consider the integral
 Here, the function is nested inside another elementary function, the sine function. Because the derivative of is just a constant, we don't need to worry about introducing any unnecessary variables. Therefore, make the substitution

2Find du. Take the derivative of u with respect to x, and solve for du.
 As you improve your technique, you will eventually jump straight to the differential instead of solving for it.
Advertisement 
3Rewrite your integral in terms of u.
 Here, we wrote the integral using du by solving for dx and replacing it. This is why there is an extra 1/2 term (which we can factor out).
 If you are left with a variable which is not u after replacing anything that you can with u and du, sometimes solving for that variable in terms of u and replacing it works. This is called backsubstitution, and the supplementary example below will use such a substitution.

4Integrate.

5Write your answer in terms of your original variable. Replace u with what you set it equivalent to earlier.
 As we can see, usubstitution is just the analogue of the chain rule from differential calculus.
Advertisement
Part 2
Part 2 of 3:Definite Integral
Download Article
Part 2

1Determine what you will use as u. This example demonstrates usubstitution of definite integrals and trigonometric functions.
 Consider the integral
 Notice that this function does not have a nested function within another function that we can use. If we consider this as a sine function cubed, the resulting usub will get us nowhere. However, using the trigonometric identity we can rewrite the integrand as
 Recall that Remember that in general, we want u so that its differential ends up canceling out part of the integrand. In this case, the
 Therefore, make the substitution

2Find du. Take the derivative of u, and solve for du.
 From above,

3Rewrite your integral so that you can express it in terms of u. Make sure to change your boundaries as well, since you changed variables. To do so, simply substitute the boundaries into your usubstitution equation.

4The extra neatly cancels out, but note the negative sign. Now, recognize that swapping the boundaries negates the integral, so we end up with a positive integral in the end.

5Integrate.
 The integrand is an even function, and the boundaries are symmetric. Therefore, we can factor out a 2 and set the lower boundary to 0 to simplify calculations.
 We did not need to do this simplification to get the correct answer, but for more complicated integrals, this technique is useful to prevent arithmetic mistakes.
 Notice that we did not rewrite our integral in terms of the original variable. Since we changed our boundaries, the integrals are equivalent. Ultimately, the objective is to solve the problem in the easiest and most efficient way possible, so there's no need to spend more time on an extra step.
Advertisement
Part 3
Part 3 of 3:Supplementary Example
Download Article
Part 3

1Evaluate the following integral. This is a more advanced example that incorporates usubstitution. In part 1, recall that we said that an integral after performing a usub may not cancel the original variables, so solving for the variable in terms of and substituting may be required. That will also be necessary in this problem.
 We see that the derivative is not If we try to immediately usub, we will end up with an increasingly complicated expression, because solving for in terms of will wind up with a square root.

2Rewrite the numerator by completing the square. Notice that the numerator just requires a to complete the square. If we just add and then subtract i.e. add 0, then we can reduce the problem to a more manageable one after simplifying.
 It is worth noting that this technique of adding 0 is a very useful one, especially in the context of completing the square. Since 0 is the additive identity, we have not actually changed the integral.

3Make the usub . The integral in the last line above is perhaps the simplest type of expression where this sort of "back substitution" is required  that is, solving for in terms of and plugging that in as well since the usub didn't cancel all the terms. Remember to change your boundaries.

4Evaluate.Advertisement
Community Q&A
Ask a Question
200 characters left
Include your email address to get a message when this question is answered.
Submit
Advertisement
Video
Tips
 Check your answer. Find the derivative of your answer and see if you recover your original function. On homework problems, use a calculator or WolframAlpha.Thanks!
 Don't forget to add the constant to the end of indefinite integrals!Thanks!
Submit a Tip
All tip submissions are carefully reviewed before being published
Thanks for submitting a tip for review!
Advertisement
About This Article
Thanks to all authors for creating a page that has been read 16,965 times.
Did this article help you?
Advertisement