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

The integral of e^{3x^2}2x is 1/3 e^{3x^2}+C

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integral of e^{3x^2}2x

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

\frac{1}{3} e^{3 x^{2}} + C

Solution steps

\int e^{3 x^{2}} \cdot 2 x d x

Take the constant out:

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

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

Apply u-substitution

: \int \frac{e ^{u}}{6} d u

= 2 \cdot \int \frac{e ^{u}}{6} d u

Take the constant out:

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

= 2 \cdot \frac{1}{6} \cdot \int e^{u} d u

Use the common integral:

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

= 2 \cdot \frac{1}{6} e^{u}

Substitute back

u = 3 x^{2}

= 2 \cdot \frac{1}{6} e^{3 x^{2}}

Simplify

2 \cdot \frac{1}{6} e^{3 x^{2}} : \frac{1}{3} e^{3 x^{2}}

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

Add a constant to the solution

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

\int x e^{- 0.2 x} d x

\int \frac{2 x}{x ^{2} + 4 x + 3} d x

\int \frac{2 x}{( x ^{2} + 2 ) ^{2}} d x

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

\int \frac{x ^{3}}{2 + x ^{2}} d x

What is the integral of e^{3x^2}2x ?
The integral of e^{3x^2}2x is 1/3 e^{3x^2}+C