What is the integral from 0 to x of sin(x/y) ?

The integral from 0 to x of sin(x/y) is y(-cos(x/y)+1)

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integral from 0 to x of sin(x/y)

\int_{0}^{x} sin (\frac{x}{y}) d x

y (- cos (\frac{x}{y}) + 1)

Solution steps

\int_{0}^{x} sin (\frac{x}{y}) d x

Apply u-substitution

: \int_{0}^{\frac{x}{y}} sin (u) y d u

= \int_{0}^{\frac{x}{y}} sin (u) y d u

Take the constant out:

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

= y \cdot \int_{0}^{\frac{x}{y}} sin (u) d u

Use the common integral:

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

= y [- cos (u)]_{0}^{\frac{x}{y}}

Compute the boundaries:

- cos (\frac{x}{y}) + 1

= y (- cos (\frac{x}{y}) + 1)

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What is the integral from 0 to x of sin(x/y) ?
The integral from 0 to x of sin(x/y) is y(-cos(x/y)+1)