What is the integral of y^2e^{xy^2} ?

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

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integral of y^2e^{xy^2}

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = y^{2} x

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

Simplify

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

= e^{y^{2} x}

Add a constant to the solution

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

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