What is the integral of 2/3 x^{3/2} 1/x ?

The integral of 2/3 x^{3/2} 1/x is 4/9 x^{3/2}+C

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integral of 2/3 x^{3/2} 1/x

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

\frac{4}{9} x^{\frac{3}{2}} + C

Solution steps

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

Take the constant out:

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

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

Simplify

x^{\frac{3}{2}} \frac{1}{x} : x^{\frac{1}{2}}

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

Apply the Power Rule

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

Simplify

\frac{2}{3} \cdot \frac{2}{3} x^{\frac{3}{2}} : \frac{4}{9} x^{\frac{3}{2}}

= \frac{4}{9} x^{\frac{3}{2}}

Add a constant to the solution

= \frac{4}{9} x^{\frac{3}{2}} + C

What is the integral of 2/3 x^{3/2} 1/x ?
The integral of 2/3 x^{3/2} 1/x is 4/9 x^{3/2}+C