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

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

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integral of xe^{x^2}

\int x 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

\int \frac{2 x ^{2} - 5 x - 2}{( x - 2 ) ^{2} ( x - 1 )} d x

f (x) = 5 x + 30, at x = - 1

x \to - 1 + lim (∣ x ∣ - 3)

\int_{- 1}^{1} (x^{100}) d x

\frac{d}{d x} (3 x^{2})