Evaluate the integral

Question

Evaluate the integral  .

Question options:

1)	2ln|1 - sin2(x)| + C
2)	-csc(x) + C
3)	sec(x) + C
4)	None of these

Answer 1

To find the antiderivative, first replace the denominator (using the Pythagorean Identity sin²(x) + cos²(x) = 1) with cos²(x). Then do a "u" substitution by letting u = cos(x). Then du = -sin(x) dx or dx = du/-sin(x). The integral becomes:

The sin(x) on top and bottom cancel and you can bring the negative out front to get

-∫u^-2 du and the antiderivative of that is u^-1 or 1/u. Now replace u with cos(x) to get 1/cos(x) + C = sec(x) + C