∫^ u
n du = + CSuppose you have to integrate ∫ (x − 4)^10 dx. You could expand out this function and integrate each term,
but that’ll take a while. Instead, you can follow these four steps.
Step 1: Let u = x − 4. Then = 1 (rearrange this to get du = dx).
Step 2: Substitute u = x − 4 and du = dx into the integrand.
∫^ u
(^10) du
Step 3: Integrate.
∫^ u
(^10) du = + C
Step 4: Substitute back for u.
+ C
That’s u-substitution. The main difficulty you’ll have will be picking the appropriate function to set equal
to u. The best way to get better is to practice. The object is to pick a function and replace it with u, then
take the derivative of u to find du. If we can’t replace all of the terms in the integrand, we can’t do the
substitution.
Let’s do some examples.
Example 1: (^) ∫ 10 x(5x^2 − 3)^6 dx =
Once again, you could expand this out and integrate each term, but that would be difficult. Use u-
substitution.
Let u = 5x^2 − 3. Then = 10x and du = 10x dx. Now you can substitute.
∫^ u
(^6) du