Question

Question #0d230

Answer 1

Depending on interval of integration, I get `+-1/(2(1-cosx)^2) +C`

Explanation:

`int (-sqrt(1+cosx))/(1-cosx)^(5/2) dx = int (-sqrt(1+cosx)(sqrt(1-cosx)))/((1-cosx)^(5/2)(1-cosx)^(1/2)) dx`

`= int(-sqrt(1-cos^2x))/(1-cosx)^3 dx`

`= int(-sqrt(sin^2x))/(1-cosx)^3 dx`

`= int(+-sinx)/(1-cosx)^3 dx`

Now integrate by substitution `u = 1-cosx` to get

`= +-1/(2(1 - cosx)^2) +C`

Note

The integrand is never positive, so the integral should be non-positive on any interval on which it exists as well.

So the indefinite integral should be:

`(-1)/(2(1 - cosx)^2) +C` `" "` (I think.)