What is the integral from-1 to 3 of xe^{xy} ?

The integral from-1 to 3 of xe^{xy} is e^{3x}-e^{-x}

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integral from-1 to 3 of xe^{xy}

\int_{- 1}^{3} x e^{x y} d y

e^{3 x} - e^{- x}

Solution steps

\int_{- 1}^{3} x e^{x y} d y

Take the constant out:

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

= x \cdot \int_{- 1}^{3} e^{x y} d y

Apply u-substitution

: \int_{- x}^{x \cdot 3} \frac{e ^{u}}{x} d u

= x \cdot \int_{- x}^{x \cdot 3} \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 \frac{1}{x} \cdot \int_{- x}^{x \cdot 3} e^{u} d u

Use the common integral:

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

= x \frac{1}{x} [e^{u}]_{- x}^{x \cdot 3}

Simplify

x \frac{1}{x} [e^{u}]_{- x}^{x \cdot 3} : [e^{u}]_{- x}^{3 x}

= [e^{u}]_{- x}^{3 x}

Compute the boundaries:

e^{3 x} - e^{- x}

= e^{3 x} - e^{- x}

- x + 4, 2 x, x

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What is the integral from-1 to 3 of xe^{xy} ?
The integral from-1 to 3 of xe^{xy} is e^{3x}-e^{-x}