able to do that step on your own.
Example 3: (^) ∫ 3 sin(3x − 1) dx =
Let u = 3x − 1. Then du = 3 dx. Substitute the u in the integral.
∫ sin u du
Figure out the integral.
∫ sin u du = −cos u + C
And throw the x’s back in.
− cos(3x − 1) + CSo far, this is only the simplest kind of u-substitution; naturally, the process can get worse when the
substitution isn’t as easy. Usually, you’ll have to insert a constant term to put the integrand into a workable
form.
Example 4: (^) ∫(5x + 7)^20 dx =
Let u = 5x + 7. Then du = 5 dx. Notice that we can’t do the substitution immediately because we need to
substitute for dx and we have 5 dx. No problem: Because 5 is a constant, just solve for dx.
du = dx
Now you can substitute.
∫ (5x + 7)
(^20) dx =
∫^ u
(^20) du
Rearrange the integral and solve.
And now it’s time to substitute back.