Cracking The Ap Calculus ab Exam 2018

Example 5: (^) ∫ x cos(3x^2 + 1) dx =
Let u = 3x^2 + 1. Then du = 6x dx. We need to substitute for x dx, so we can rearrange the du term.
du = x dx
Now substitute.

∫^ cos u du

Evaluate the integral.

∫^ cos u du = sin u + C

And substitute back.

    sin u   +   C   =       sin(3x^2    +   1)  +   C

Example 6: (^) ∫ x sec^2 (x^2 ) dx =
Let u = x^2 . Then du = 2x dx and du = x dx.
Substitute.

∫^ sec

(^2) u du
Evaluate the integral.

∫ sec

(^2) u du = tan u + C
Now the original function goes back in.
tan (x^2 ) + C