What is the integral of 4xye^{xy^2} ?

The integral of 4xye^{xy^2} is 2e^{xy^2}+C

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integral of 4xye^{xy^2}

\int 4 x y e^{x y^{2}} d y

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

Solution steps

\int 4 x y e^{x y^{2}} d y

Take the constant out:

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

= 4 x \cdot \int e^{x y^{2}} y d y

Apply u-substitution

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

= 4 x \cdot \int \frac{e ^{u}}{2 x} d u

Take the constant out:

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

= 4 x \frac{1}{2 x} \cdot \int e^{u} d u

Use the common integral:

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

= 4 x \frac{1}{2 x} e^{u}

Substitute back

u = x y^{2}

= 4 x \frac{1}{2 x} e^{x y^{2}}

Simplify

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

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

Add a constant to the solution

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

f (x) = x^{4} - 20 x^{2} + 64, at x = - 2

24 x^{3} + 24 x^{2}

\frac{d}{d x} (2 x^{2} - \frac{1}{x ^{2}} + \frac{1}{x ^{3}})

\int \frac{31}{x ^{3} - 27} d x

f (x) = 7 x + 5

What is the integral of 4xye^{xy^2} ?
The integral of 4xye^{xy^2} is 2e^{xy^2}+C