What is the integral of (sqrt(x^3))/3 ?

The integral of (sqrt(x^3))/3 is 2/15 x^{5/2}+C

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integral of (sqrt(x^3))/3

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

\frac{2}{15} x^{\frac{5}{2}} + C

Solution steps

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

Take the constant out:

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

= \frac{1}{3} \cdot \int x^{3} d x

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

assuming

x \geq 0

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

Apply the Power Rule

= \frac{1}{3} \cdot \frac{2}{5} x^{\frac{5}{2}}

Simplify

\frac{1}{3} \cdot \frac{2}{5} x^{\frac{5}{2}} : \frac{2}{15} x^{\frac{5}{2}}

= \frac{2}{15} x^{\frac{5}{2}}

Add a constant to the solution

= \frac{2}{15} x^{\frac{5}{2}} + C

What is the integral of (sqrt(x^3))/3 ?
The integral of (sqrt(x^3))/3 is 2/15 x^{5/2}+C