How do you integrate #int (sinx) / ((cos^2x + cosx -2)) dx# using partial fractions?
When dealing with trigonometric functions during partial fraction expansions, it can be very helpful to make some substitutions to simplify the problem. We will let
We may actually take this one step further and eliminate the
One can think of this as being equivalent to
Now we can simply expand our new, trig-less expression and integrate normally.
We will factor the denominator, and set the expression equal to a general expression involving these factors to begin the partial fraction expansion:
To continue our expansion we will multiply through by
We may now find
On the left-hand side, we have no multiples of
This is a simple linear system which I will not bother with in too much detail, so you'll just have to trust me when I say the solution is
The expanded integral, then, is
From here the answer should be fairly clear:
However, we must substitute
After a small tweak for the sake of aesthetics: