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

The integral of sin(y/x) is -xcos(y/x)+C

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integral of sin(y/x)

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

- x cos (\frac{y}{x}) + C

Solution steps

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

Apply u-substitution

: \int sin (u) x d u

= \int sin (u) x d u

Take the constant out:

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

= x \cdot \int sin (u) d u

Use the common integral:

\int sin (u) d u = - cos (u)

= x (- cos (u))

Substitute back

u = \frac{y}{x}

= x (- cos (\frac{y}{x}))

Simplify

= - x cos (\frac{y}{x})

Add a constant to the solution

= - x cos (\frac{y}{x}) + C

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