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

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

Popular >

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

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

\frac{2}{3} x^{\frac{3}{2}} - \frac{2}{5} x^{\frac{5}{2}} + C

Solution steps

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

Expand

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

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

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

= \frac{2}{3} x^{\frac{3}{2}} - \frac{2}{5} x^{\frac{5}{2}}

Add a constant to the solution

= \frac{2}{3} x^{\frac{3}{2}} - \frac{2}{5} x^{\frac{5}{2}} + C

\int \frac{5 x - 15}{( x ^{2} + 2 x + 4 ) ^{2}} d x

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

\int x (1 + x)^{4} d x

\int \frac{3 x + 5}{x ^{2} - 3 x + 2} d x

\int \frac{x ^{2} + 1}{x ^{2} - 4} - x d x

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