What is the integral of 1/3 sqrt(x) ?

The integral of 1/3 sqrt(x) is 2/9 x^{3/2}+C

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integral of 1/3 sqrt(x)

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

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

Solution steps

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

Apply the Power Rule

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

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

Simplify

\frac{1}{3} \cdot \frac{2}{3} x^{\frac{3}{2}} : \frac{2}{9} x^{\frac{3}{2}}

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

Add a constant to the solution

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

y = x^{2} sin (2 x)

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

\frac{d}{d x} (x^{3} + 3 x^{2} + 3 x + 1)

\frac{\partial}{\partial x} (3^{2 x} + 5^{4 y} + 2 x + y^{4})

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

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