What is the integral of 3y^3e^{xy} ?

The integral of 3y^3e^{xy} is 3y^2e^{yx}+C

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integral of 3y^3e^{xy}

\int 3 y^{3} e^{x y} d x

3 y^{2} e^{y x} + C

Solution steps

\int 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 e^{y x} d x

Apply u-substitution

= 3 y^{3} \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

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

Use the common integral:

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

= 3 y^{3} \frac{1}{y} e^{u}

Substitute back

u = y x

= 3 y^{3} \frac{1}{y} e^{y x}

Simplify

3 y^{3} \frac{1}{y} e^{y x} : 3 y^{2} e^{y x}

= 3 y^{2} e^{y x}

Add a constant to the solution

= 3 y^{2} e^{y x} + C