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

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

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integral of sqrt(x)e^{sqrt(x^3)}

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

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

Solution steps

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

x^{3} = x^{\frac{3}{2}},

assuming

x \geq 0

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

Apply u-substitution

= \int 2 e^{u^{3}} u^{2} d u

Take the constant out:

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

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

Apply u-substitution

= 2 \cdot \int \frac{e ^{v}}{3} d v

Take the constant out:

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

= 2 \cdot \frac{1}{3} \cdot \int e^{v} d v

Use the common integral:

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

= 2 \cdot \frac{1}{3} e^{v}

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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