What is the integral of 4axe^{ax^2} ?

The integral of 4axe^{ax^2} is 2e^{ax^2}+C

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integral of 4axe^{ax^2}

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = a x^{2}

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

Simplify

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

= 2 e^{a x^{2}}

Add a constant to the solution

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

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What is the integral of 4axe^{ax^2} ?
The integral of 4axe^{ax^2} is 2e^{ax^2}+C