Question #0d230

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Answer 1 · 2017-08-05T00:25:23

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

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

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

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

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

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.)