What is the integral from 0 to y^2 of 3y^3e^{xy} ?

The integral from 0 to y^2 of 3y^3e^{xy} is 3y^2(e^{y^3}-1)

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integral from 0 to y^2 of 3y^3e^{xy}

\int_{0}^{y^{2}} 3 y^{3} e^{x y} d x

3 y^{2} (e^{y^{3}} - 1)

Solution steps

\int_{0}^{y^{2}} 3 y^{3} e^{x y} d x

Take the constant out:

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

= 3 y^{3} \cdot \int_{0}^{y^{2}} e^{y x} d x

Apply u-substitution

: \int_{0}^{y^{3}} \frac{e ^{u}}{y} d u

= 3 y^{3} \cdot \int_{0}^{y^{3}} \frac{e ^{u}}{y} d u

Take the constant out:

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

= 3 y^{3} \frac{1}{y} \cdot \int_{0}^{y^{3}} e^{u} d u

Use the common integral:

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

= 3 y^{3} \frac{1}{y} [e^{u}]_{0}^{y^{3}}

Simplify

3 y^{3} \frac{1}{y} [e^{u}]_{0}^{y^{3}} : 3 y^{2} [e^{u}]_{0}^{y^{3}}

= 3 y^{2} [e^{u}]_{0}^{y^{3}}

Compute the boundaries:

e^{y^{3}} - 1

= 3 y^{2} (e^{y^{3}} - 1)

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What is the integral from 0 to y^2 of 3y^3e^{xy} ?
The integral from 0 to y^2 of 3y^3e^{xy} is 3y^2(e^{y^3}-1)