What is the integral of sin(pi/x) ?

The integral of sin(pi/x) is xsin(pi/x)-pi\Ci(pi/x)+C

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integral of sin(pi/x)

\int sin (\frac{π}{x}) d x

x sin (\frac{π}{x}) - π Ci (\frac{π}{x}) + C

Solution steps

\int sin (\frac{π}{x}) d x

Apply u-substitution

: \int - \frac{π sin ( u )}{u ^{2}} d u

= \int - \frac{π sin ( u )}{u ^{2}} d u

Take the constant out:

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

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

Apply Integration By Parts

: - \frac{sin ( u )}{u} - \int - \frac{cos ( u )}{u} d u

= - π (- \frac{sin ( u )}{u} - \int - \frac{cos ( u )}{u} d u)

\int - \frac{cos ( u )}{u} d u = - Ci (u)

= - π (- \frac{sin ( u )}{u} - (- Ci (u)))

Substitute back

u = \frac{π}{x}

= - π (- \frac{sin ( \frac{π}{x} )}{\frac{π}{x}} - (- Ci (\frac{π}{x})))

Simplify

- π (- \frac{sin ( \frac{π}{x} )}{\frac{π}{x}} - (- Ci (\frac{π}{x}))) : x sin (\frac{π}{x}) - π Ci (\frac{π}{x})

= x sin (\frac{π}{x}) - π Ci (\frac{π}{x})

Add a constant to the solution

= x sin (\frac{π}{x}) - π Ci (\frac{π}{x}) + C

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What is the integral of sin(pi/x) ?
The integral of sin(pi/x) is xsin(pi/x)-pi\Ci(pi/x)+C