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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

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

Simplify

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

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

Compute the boundaries:

e^{x^{2}} - 1

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

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