What is the integral of csc(pix)cos(pix) ?

The integral of csc(pix)cos(pix) is xcot(pix)-(-pi 1/(pi^2)(-pixcot(pix)-(-ln|sin(pix)|)))+C

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integral of csc(pix)cos(pix)

\int csc (π x) cos (π x) d x

x cot (π x) - (- π \frac{1}{π ^{2}} (- π x cot (π x) - (- ln ∣ sin (π x) ∣))) + C

Solution steps

\int csc (π x) cos (π x) d x

Simplify

csc (π x) cos (π x) : cot (π x)

= \int cot (π x) d x

Apply Integration By Parts

= x cot (π x) - \int - π x csc^{2} (π x) d x

\int - π x csc^{2} (π x) d x = - π \frac{1}{π ^{2}} (- π x cot (π x) - (- ln ∣ sin (π x) ∣))

= x cot (π x) - (- π \frac{1}{π ^{2}} (- π x cot (π x) - (- ln ∣ sin (π x) ∣)))

Add a constant to the solution

= x cot (π x) - (- π \frac{1}{π ^{2}} (- π x cot (π x) - (- ln ∣ sin (π x) ∣))) + C

What is the integral of csc(pix)cos(pix) ?
The integral of csc(pix)cos(pix) is xcot(pix)-(-pi 1/(pi^2)(-pixcot(pix)-(-ln|sin(pix)|)))+C