What is the integral of xye^{yz} ?

The integral of xye^{yz} is xe^{yz}+C

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integral of xye^{yz}

\int x y e^{yz} d z

x e^{yz} + C

Solution steps

\int x y e^{yz} d z

Take the constant out:

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

= x y \cdot \int e^{yz} d z

Apply u-substitution

= x y \cdot \int \frac{e ^{u}}{y} d u

Take the constant out:

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

= x y \frac{1}{y} \cdot \int e^{u} d u

Use the common integral:

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

= x y \frac{1}{y} e^{u}

Substitute back

u = yz

= x y \frac{1}{y} e^{yz}

Simplify

x y \frac{1}{y} e^{yz} : x e^{yz}

= x e^{yz}

Add a constant to the solution

= x e^{yz} + C