# When do you use u-substitution in integration?

## Main Question or Discussion Point

When do you use "u-substitution" in integration?

I understand how to use it, but I'm just getting really confused on when you use it? Is there a way you can look at the problem and tell you need substitution vs. other methods? We have a final coming up, and obviously we won't have too much time to think so any suggestions are appreciated! Thank you!

Maylis
Gold Member
I think a good indicator that you should use it is if you have a function inside the integral and if you know its derivative, you see the derivative as well.

Take for example

∫sin(x)cos(x)dx.

You probably know that the derivative of sin(x) is cos(x)dx. So in that particular case its easy to call sin(x) u, then du is just the derivative of u, so then your integral becomes ∫u du, which is much more clean and easy to solve.

Mark44
Mentor
Substitution is among the simpler techniques that can be used, as compared to integration by parts and trig substitutions. I find that it is good to try out the simple substitutions before going on to the more complicated techniques. Being more complicated, they tend to take more time and offer more opportunities for errors.

Generally, the rule of thumb for integrating by substitution is having an integrand such that the argument of the more complex function is of the same "family" as the less complex function. For example, take:
$\int e^{x^2} \ 2x dx$
In this example, the more complex function is the exponential, thus, we will check to see if the argument of the exponential is the same as the other function. The argument of $e^{x^2}$ is $x^2$, thus $\frac{d}{dx} x^2 = 2x$, which is a polynomial like the less complex function. On another note, if you have a rational function (a ratio of two polynomials), nine times out of ten, $u$ will be the denominator.
When the argument of the more complex function is not of the same class as the other function, such as
$\int e^{x^2} sin(x) dx$
you will have to integrate by parts.

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You use the "u-substituion" when you think that this change of variable will facilitate the computation for you.

For example: compute the integral $\int \sqrt{1-x^2}dx$

Well, I know that $\cos(\theta) = \sqrt{1-\sin(\theta)^2}$, so if I say that $x = \sin(\theta)$ thus

$\int \sqrt{1-x^2}dx = \int \sqrt{1-\sin(\theta)^2} d\sin(\theta) = \int \sqrt{1-\sin(\theta)^2} \frac{d\sin(\theta)}{d\theta}d\theta = \int \cos(\theta) \cos(\theta) d\theta= \int \cos(\theta)^2 d\theta$ and this last integral is easy of compute