integration help!!!!!

Question

∫cosxdx/(sinx+sin^2x)^2

Answer 1

Let u = sinx

Dervative with respect to x

 du = cosxdx

In the denominator add and subtract to 1/4

Apply the formula: .

Let s = 1/2 +u

Dervative with respect to u

ds = du

Apply partial fraction:

 

Let p = -1+2s                            v = 1+2s

Dervative with respect to s          Dervative with respect to s

dp = 2ds                                   dv = 2ds

ds = dp/2                                  ds = dv/2

Substitute the above values

Integrate the sum term by term and factor out constants

Substitute in  p and v values

Substitute in s value

Substitute in u value

The solution is 

 

 

 

Answer 2

solution to the probem is