What is the integral of xe^{2x^2} ?

The integral of xe^{2x^2} is 1/4 e^{2x^2}+C

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integral of xe^{2x^2}

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

\frac{1}{4} e^{2 x^{2}} + C

Solution steps

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

Apply u-substitution

: \int \frac{e ^{u}}{4} d u

= \int \frac{e ^{u}}{4} d u

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = 2 x^{2}

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

Add a constant to the solution

= \frac{1}{4} e^{2 x^{2}} + C

\frac{\partial}{\partial y} (x + 3 x^{3} sin (y))

\frac{d y}{d x} = \frac{y ^{3}}{x}

\int 8 + tan^{2} (x) d x

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\frac{d y}{d x} = (\frac{( 4 y + 3 )}{8 x + 5})^{2}