What is the integral of x*e^{x^2} ?

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

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integral of x*e^{x^2}

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

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

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = x^{2}

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

Add a constant to the solution

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

\frac{d ^{2} x}{d t ^{2}} + 9 x = - sin (3 t)

\frac{\partial}{\partial x} (ln (\frac{2 x y}{z}))

\int \frac{5 x}{3 x ^{\frac{1}{3}}} d x

y = 1.1 x + e^{x}, (0, 1)