What is the integral of yx^3e^{x^2y^2} ?

The integral of yx^3e^{x^2y^2} is (xe^{x^2y^2})/2+C

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integral of yx^3e^{x^2y^2}

\int y x^{3} e^{x^{2} y^{2}} d y

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

Solution steps

\int y x^{3} e^{x^{2} y^{2}} d y

Take the constant out:

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

= x^{3} \cdot \int e^{x^{2} y^{2}} y d y

Apply exponent rule:

a^{n} \cdot b^{n} = (a \cdot b)^{n}

x^{2} y^{2} = (x y)^{2}

= x^{3} \cdot \int e^{(x y)^{2}} y d y

Apply u-substitution

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

= x^{3} \cdot \int \frac{e ^{u^{2}} u}{x ^{2}} d u

Take the constant out:

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

= x^{3} \frac{1}{x ^{2}} \cdot \int e^{u^{2}} u d u

Apply u-substitution

: \int \frac{e ^{v}}{2} d v

= x^{3} \frac{1}{x ^{2}} \cdot \int \frac{e ^{v}}{2} d v

Take the constant out:

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

= x^{3} \frac{1}{x ^{2}} \cdot \frac{1}{2} \cdot \int e^{v} d v

Use the common integral:

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

= x^{3} \frac{1}{x ^{2}} \cdot \frac{1}{2} e^{v}

Substitute back

= x^{3} \frac{1}{x ^{2}} \cdot \frac{1}{2} e^{(x y)^{2}}

Simplify

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

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

Add a constant to the solution

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

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What is the integral of yx^3e^{x^2y^2} ?
The integral of yx^3e^{x^2y^2} is (xe^{x^2y^2})/2+C