What is the integral of (csc^2(pi/x))/(x^2) ?

The integral of (csc^2(pi/x))/(x^2) is 1/pi cot(pi/x)+C

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integral of (csc^2(pi/x))/(x^2)

\int \frac{csc ^{2} ( \frac{π}{x} )}{x ^{2}} d x

\frac{1}{π} cot (\frac{π}{x}) + C

Solution steps

\int \frac{csc ^{2} ( \frac{π}{x} )}{x ^{2}} d x

Apply u-substitution

: \int - \frac{csc ^{2} ( u )}{π} d u

= \int - \frac{csc ^{2} ( u )}{π} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \frac{1}{π} \cdot \int csc^{2} (u) d u

Use the common integral:

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

= - \frac{1}{π} (- cot (u))

Substitute back

u = \frac{π}{x}

= - \frac{1}{π} (- cot (\frac{π}{x}))

Simplify

= \frac{1}{π} cot (\frac{π}{x})

Add a constant to the solution

= \frac{1}{π} cot (\frac{π}{x}) + C

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What is the integral of (csc^2(pi/x))/(x^2) ?
The integral of (csc^2(pi/x))/(x^2) is 1/pi cot(pi/x)+C