What is the integral of 10xe^{x^2} ?

The integral of 10xe^{x^2} is 5e^{x^2}+C

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integral of 10xe^{x^2}

\int 10 x e^{x^{2}} d x

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

Solution steps

\int 10 x e^{x^{2}} d x

Take the constant out:

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

= 10 \cdot \int x e^{x^{2}} d x

Apply u-substitution

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

= 10 \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

= 10 \cdot \frac{1}{2} \cdot \int e^{u} d u

Use the common integral:

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

= 10 \cdot \frac{1}{2} e^{u}

Substitute back

u = x^{2}

= 10 \cdot \frac{1}{2} e^{x^{2}}

Simplify

10 \cdot \frac{1}{2} e^{x^{2}} : 5 e^{x^{2}}

= 5 e^{x^{2}}

Add a constant to the solution

= 5 e^{x^{2}} + C

f (x) = 1 + 2 e^{3 x}

8 (x - \frac{1}{x})^{2}

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