What is the integral of 2xye^{x^2y} ?

The integral of 2xye^{x^2y} is e^{yx^2}+C

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integral of 2xye^{x^2y}

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{e ^{u}}{2 y} d u

= 2 y \cdot \int \frac{e ^{u}}{2 y} d u

Take the constant out:

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

= 2 y \frac{1}{2 y} \cdot \int e^{u} d u

Use the common integral:

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

= 2 y \frac{1}{2 y} e^{u}

Substitute back

u = y x^{2}

= 2 y \frac{1}{2 y} e^{y x^{2}}

Simplify

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

= e^{y x^{2}}

Add a constant to the solution

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

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