What is the integral of (x^3-2x)/(sqrt(x)) ?

The integral of (x^3-2x)/(sqrt(x)) is 2/7 x^{7/2}-4/3 x^{3/2}+C

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integral of (x^3-2x)/(sqrt(x))

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

\frac{2}{7} x^{\frac{7}{2}} - \frac{4}{3} x^{\frac{3}{2}} + C

Solution steps

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

Simplify

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

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

Expand

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

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

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

\int 2 x d x = \frac{4}{3} x^{\frac{3}{2}}

= \frac{2}{7} x^{\frac{7}{2}} - \frac{4}{3} x^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{7} x^{\frac{7}{2}} - \frac{4}{3} x^{\frac{3}{2}} + C

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What is the integral of (x^3-2x)/(sqrt(x)) ?
The integral of (x^3-2x)/(sqrt(x)) is 2/7 x^{7/2}-4/3 x^{3/2}+C