What is the integral of (2x)(e^{x^2+3}) ?

The integral of (2x)(e^{x^2+3}) is e^{x^2+3}+C

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integral of (2x)(e^{x^2+3})

\int (2 x) (e^{x^{2} + 3}) d x

e^{x^{2} + 3} + C

Solution steps

\int 2 x e^{x^{2} + 3} 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} + 3} 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} + 3

= 2 \cdot \frac{1}{2} e^{x^{2} + 3}

Simplify

2 \cdot \frac{1}{2} e^{x^{2} + 3} : e^{x^{2} + 3}

= e^{x^{2} + 3}

Add a constant to the solution

= e^{x^{2} + 3} + C

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What is the integral of (2x)(e^{x^2+3}) ?
The integral of (2x)(e^{x^2+3}) is e^{x^2+3}+C