intx -csc(x) dx
=int x dx - int csc(x) dx
=(x^2)/2-ln(abs(csc(x) - cot(x))) + C
Where C is a constant of integration.
So f(x) = (x^2)/2-ln(abs(csc(x) - cot(x))) + C
Using the fact that f((5pi)/4) = 0 , we can substitute:
f((5pi)/4)=(((5pi)/4)^2)/2-ln(abs(csc((5pi)/4) - cot((5pi)/4))) + C
Moreover,
(((5pi)/4)^2)/2-ln(abs(csc((5pi)/4) - cot((5pi)/4))) + C = 0
Simplifying, we get
(25pi^2)/32-ln(abs(-sqrt(2)-1))+C=0
(25pi^2)/32-ln(sqrt(2)+1)+C=0
C=ln(sqrt(2)+1)-(25pi^2)/32
Therefore, the answer is:
f(x) = (x^2)/2-ln(abs(csc(x) - cot(x))) + ln(sqrt(2)+1)-(25pi^2)/32
Here's an image of what the graph looks like:
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