x) dx. Now, we can evaluate the integral: ∫(x − 2 sec^2 x) dx = − 2 tan x + C.
SOLUTIONS TO PRACTICE PROBLEM SET 19
1.
If we let u = sin 2x, then du = 2 cos 2x dx. We need to substitute for cos 2x dx, so we can
divide the du term by 2: = cos 2x dx. Next we can substitute into the integral: ∫sin 2x cos
2 x dx = ∫u du. Now we can integrate: . Last, we
substitute back and get + C.
2.
First, pull the constant out of the integrand: dx = 3 dx. If we let u =
10 − x^2 , then du = −2x dx. We need to substitute for x dx, so we can divide the du term by −2: =
= x dx. Next we can substitute into the integral: . Now
we can integrate: = = . Last, we substitute back and get
.
3.
If we let u = 5x^4 + 20, then du = 20x^3 dx. We need to substitute for x^3 dx, so we can divide the
du term by 20: = x^3 dx. Next we can substitute into the integral:
