What is the integral of xe^{y+2z} ?

The integral of xe^{y+2z} is xe^{2z+y}+C

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integral of xe^{y+2z}

\int x e^{y + 2 z} d y

x e^{2 z + y} + C

Solution steps

\int x e^{y + 2 z} d y

Take the constant out:

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

= x \cdot \int e^{y + 2 z} d y

Simplify

e^{y + 2 z} : e^{y} e^{2 z}

= x \cdot \int e^{y} e^{2 z} d y

Take the constant out:

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

= x e^{2 z} \cdot \int e^{y} d y

Use the common integral:

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

= x e^{2 z} e^{y}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

e^{2 z} e^{y} = e^{2 z + y}

= x e^{2 z + y}

Add a constant to the solution

= x e^{2 z + y} + C