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

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

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integral of (2xe^{x^2+3y})

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

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

Solution steps

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

Take the constant out:

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

= 2 \cdot \int e^{x^{2} + 3 y} x d x

Apply u-substitution

= 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 y

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

Simplify

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

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

Add a constant to the solution

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

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