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

The integral of (x^3-x)/(sqrt(x)) is 2/7 x^{7/2}-2/3 x^{3/2}+C

Popular >

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

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

\frac{2}{7} x^{\frac{7}{2}} - \frac{2}{3} x^{\frac{3}{2}} + C

Solution steps

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

Simplify

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

= \int x (x^{2} - 1) d x

Expand

x (x^{2} - 1) : x^{\frac{5}{2}} - x

= \int x^{\frac{5}{2}} - x d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{\frac{5}{2}} d x - \int x d x

\int x^{\frac{5}{2}} d x = \frac{2}{7} x^{\frac{7}{2}}

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

= \frac{2}{7} x^{\frac{7}{2}} - \frac{2}{3} x^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{7} x^{\frac{7}{2}} - \frac{2}{3} x^{\frac{3}{2}} + C

\frac{\partial}{\partial x} (x^{4} y^{9} + 6 x^{7} y)

y^{^{'}} - \frac{7}{x} \cdot y = 0

\int \frac{1}{1 - cos ( x ) + sin ( x )} d x

\frac{d}{d x} (\frac{5 x ^{4} + 7 x ^{2} - 3}{2 x ^{2}})

\int 4 x e^{x^{2}} d x

What is the integral of (x^3-x)/(sqrt(x)) ?
The integral of (x^3-x)/(sqrt(x)) is 2/7 x^{7/2}-2/3 x^{3/2}+C