What is the integral of sin(3/x) ?

The integral of sin(3/x) is xsin(3/x)-3\Ci(3/x)+C

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integral of sin(3/x)

\int sin (\frac{3}{x}) d x

x sin (\frac{3}{x}) - 3 Ci (\frac{3}{x}) + C

Solution steps

\int sin (\frac{3}{x}) d x

Apply u-substitution

: \int - \frac{3 sin ( u )}{u ^{2}} d u

= \int - \frac{3 sin ( u )}{u ^{2}} d u

Take the constant out:

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

= - 3 \cdot \int \frac{sin ( u )}{u ^{2}} d u

Apply Integration By Parts

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

= - 3 (- \frac{sin ( u )}{u} - \int - \frac{cos ( u )}{u} d u)

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

= - 3 (- \frac{sin ( u )}{u} - (- Ci (u)))

Substitute back

u = \frac{3}{x}

= - 3 (- \frac{sin ( \frac{3}{x} )}{\frac{3}{x}} - (- Ci (\frac{3}{x})))

Simplify

- 3 (- \frac{sin ( \frac{3}{x} )}{\frac{3}{x}} - (- Ci (\frac{3}{x}))) : x sin (\frac{3}{x}) - 3 Ci (\frac{3}{x})

= x sin (\frac{3}{x}) - 3 Ci (\frac{3}{x})

Add a constant to the solution

= x sin (\frac{3}{x}) - 3 Ci (\frac{3}{x}) + C

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What is the integral of sin(3/x) ?
The integral of sin(3/x) is xsin(3/x)-3\Ci(3/x)+C