18 - Advanced Integration - Integration by u-substitution

Integration is an exhausting process. Sometimes even though a function might be integrable it may be impossible or too long to compute the integral via traditional methods.

Now we'll introduce one of the most powerful rules of integration:

Integration by u-substitution

Integration by u-substitution works if you have some function of x and its derivative in THE SAME equation.

When you have such case, you transform the function into the variable u, which is a function in terms of x, and the derivative of that function into the derivative of u wrt. dx. Instead of dx or whatever variable you started with, you'll now have du. That is to say you'll integrate with respect to u.

BEWARE OF U-SUBSTITUTION IN DEFINITE INTEGRALS!

Whenever you have a definite integral, remember that the bounds are in terms of x. You'll have to transform the bounds in terms of u.
DO NOT TAKE THE BOUNDS AS THEY ARE IN THE FINAL EVALUATION. IT WILL BE WRONG.

See a familiar pattern?

Yup! The u-substitution essentially undoes chain rules, though it's more complicated than that.

After you're done integrating, you rewrite the end equation in terms of x, from terms of u.