What is #f(x) = int x-cscx dx# if #f((5pi)/4) = 0 #?

1 Answer
Jul 16, 2018

#f(x) = (x^2)/2-ln(abs(csc(x) - cot(x))) + ln(sqrt(2)+1)-(25pi^2)/32#

Explanation:

#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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