What is the integral from 0 to y of y^2e^{xy} ?

The integral from 0 to y of y^2e^{xy} is y(e^{y^2}-1)

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integral from 0 to y of y^2e^{xy}

\int_{0}^{y} y^{2} e^{x y} d x

y (e^{y^{2}} - 1)

Solution steps

\int_{0}^{y} y^{2} e^{x y} d x

Take the constant out:

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

= y^{2} \cdot \int_{0}^{y} e^{y x} d x

Apply u-substitution

: \int_{0}^{y^{2}} \frac{e ^{u}}{y} d u

= y^{2} \cdot \int_{0}^{y^{2}} \frac{e ^{u}}{y} d u

Take the constant out:

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

= y^{2} \frac{1}{y} \cdot \int_{0}^{y^{2}} e^{u} d u

Use the common integral:

\int e^{u} d u = e^{u}

= y^{2} \frac{1}{y} [e^{u}]_{0}^{y^{2}}

Simplify

y^{2} \frac{1}{y} [e^{u}]_{0}^{y^{2}} : y [e^{u}]_{0}^{y^{2}}

= y [e^{u}]_{0}^{y^{2}}

Compute the boundaries:

e^{y^{2}} - 1

= y (e^{y^{2}} - 1)

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What is the integral from 0 to y of y^2e^{xy} ?
The integral from 0 to y of y^2e^{xy} is y(e^{y^2}-1)