integration

1.

Find dx using reverse chain rule start from u = sin x

2.

Replace in the original expression: sin x to u and dx to your differential.

3.

Notice the denominator turn into cos² x.

4.

You need everything to be in terms of u to integrate.

5.

Realise you can manipulate u = sin x from the beginning to find cos² x. Hint: Square both sides and then use sin² x + cos² x.

6.

Now you should have everything in terms of u so: (1 + u) / (1 - u² ) du.

7.

Realise that the denominator is a difference of two squares: (1+ u) / (1 - u)(1 + u)

8.

(1 + u) at the top and bottom cancel out so you are left with simply: 1 / (1 - u)

9.

This integrates to ln |1 - u | + c

10.

Finally u = sin x

11.

Solution: ln |1 - sin x | + c