What is the integral of x(8e^{x^2}) ?

The integral of x(8e^{x^2}) is 4e^{x^2}+C

integral of x(8e^{x^2})

\int x (8 e^{x^{2}}) d x

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

Solution steps

\int x \cdot 8 e^{x^{2}} d x

Take the constant out:

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

= 8 \cdot \int x e^{x^{2}} d x

Apply u-substitution

= 8 \cdot \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

= 8 \cdot \frac{1}{2} \cdot \int e^{u} d u

Use the common integral:

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

= 8 \cdot \frac{1}{2} e^{u}

Substitute back

u = x^{2}

= 8 \cdot \frac{1}{2} e^{x^{2}}

Simplify

8 \cdot \frac{1}{2} e^{x^{2}} : 4 e^{x^{2}}

= 4 e^{x^{2}}

Add a constant to the solution

= 4 e^{x^{2}} + C

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