What is the integral of x4e^{x^2} ?

The integral of x4e^{x^2} is 2e^{x^2}+C

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integral of x4e^{x^2}

\int x 4 e^{x^{2}} d x

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

Solution steps

\int x \cdot 4 e^{x^{2}} d x

Take the constant out:

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

= 4 \cdot \int x e^{x^{2}} d x

Apply u-substitution

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

= 4 \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

= 4 \cdot \frac{1}{2} \cdot \int e^{u} d u

Use the common integral:

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

= 4 \cdot \frac{1}{2} e^{u}

Substitute back

u = x^{2}

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

Simplify

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

= 2 e^{x^{2}}

Add a constant to the solution

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

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