What is the integral from 0 to y^2 of ycos(x/y) ?

The integral from 0 to y^2 of ycos(x/y) is y^2sin(y)

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integral from 0 to y^2 of ycos(x/y)

\int_{0}^{y^{2}} y cos (\frac{x}{y}) d x

y^{2} sin (y)

Solution steps

\int_{0}^{y^{2}} y cos (\frac{x}{y}) d x

Take the constant out:

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

= y \cdot \int_{0}^{y^{2}} cos (\frac{x}{y}) d x

Apply u-substitution

: \int_{0}^{y} cos (u) y d u

= y \cdot \int_{0}^{y} cos (u) y d u

Take the constant out:

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

= yy \cdot \int_{0}^{y} cos (u) d u

Use the common integral:

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

= yy [sin (u)]_{0}^{y}

Simplify

yy [sin (u)]_{0}^{y} : y^{2} [sin (u)]_{0}^{y}

= y^{2} [sin (u)]_{0}^{y}

Compute the boundaries:

sin (y)

= y^{2} sin (y)

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\int_{4}^{8} \frac{3}{x ^{3}} d x

What is the integral from 0 to y^2 of ycos(x/y) ?
The integral from 0 to y^2 of ycos(x/y) is y^2sin(y)