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

The integral of u(x)e^{x^2} is u 1/2 e^{x^2}+C

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

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

u \frac{1}{2} e^{x^{2}} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

v = x^{2}

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

Add a constant to the solution

= u \frac{1}{2} e^{x^{2}} + C

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What is the integral of u(x)e^{x^2} ?
The integral of u(x)e^{x^2} is u 1/2 e^{x^2}+C