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

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

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integral of (2xe^{x^2})

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

e^{x^{2}} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

= 2 \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

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

Use the common integral:

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

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

Substitute back

u = x^{2}

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

Simplify

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

= e^{x^{2}}

Add a constant to the solution

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

y = x^{3}, y = x^{2} - 2 x, [0, 1]

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