What is the integral of: cos(x)/ (2sin (x) + 12) ?

Integral ( cos(x) / ( 2sin(x) + 12) dx )

Let me show you the substitution in action.

First, I'm going to move cos(x) next to the dx.

Integral ( 1/(2sin(x) + 12) cos(x) dx )

Now to use substitution.

Let u = 2sin(x) + 12. Then

du = 2cos(x) dx, which means

(1/2) du = cos(x) dx

The reason why I put cos(x) next to the dx is to have cos(x) dx.

Notice how cos(x) dx = (1/2) du.

That means (1/2) du will be the tail end after the substitution.

Integral ( 1/u (1/2) du )

Factor the constant out of the integral.

(1/2) Integral ( (1/u) du )

Integrate normally.

(1/2) ln|u| + C

Back-substitute u = 2sin(x) + 12.

(1/2) ln | 2sin(x) + 12 | + C

Substitute u = (2sinx + 12), so that du = 2cos x dx

The answer is ln(2sin x + 12) / 2 + C

Fib

I got the same answer as Kevin.

.5log(sin(x)+6)