Integral of x-sqrt(x) dx

The solution

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$$\int\limits_{0}^{1} \left(- \sqrt{x} + x\right)\, dx$$

Integral(x - sqrt(x), (x, 0, 1))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \sqrt{x}\right)\, dx = - \int \sqrt{x}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int \sqrt{x}\, dx = \frac{2 x^{\frac{3}{2}}}{3}$
  So, the result is: $- \frac{2 x^{\frac{3}{2}}}{3}$
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x\, dx = \frac{x^{2}}{2}$
The result is: $- \frac{2 x^{\frac{3}{2}}}{3} + \frac{x^{2}}{2}$
Add the constant of integration:

$- \frac{2 x^{\frac{3}{2}}}{3} + \frac{x^{2}}{2}+ \mathrm{constant}$

The answer is:

$- \frac{2 x^{\frac{3}{2}}}{3} + \frac{x^{2}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

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$$\int \left(- \sqrt{x} + x\right)\, dx = C - \frac{2 x^{\frac{3}{2}}}{3} + \frac{x^{2}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-1/6

$$- \frac{1}{6}$$

-1/6

$$- \frac{1}{6}$$

-1/6

Numerical answer [src]

-0.166666666666667

-0.166666666666667

Use the examples entering the upper and lower limits of integration.

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Integral of x-sqrt(x) dx

Limits of integration:

The graph:

Piecewise:

The solution