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Identical expressions
xdx/sqrt5- two x^2
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xdx/√5-2x^2
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xdx/sqrt5-2x²
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Similar expressions
xdx/sqrt5+2x^2

Integral of xdx/sqrt5-2x^2 dx

The solution

You have entered [src]

  1                  
  /                  
 |                   
 |  /  x        2\   
 |  |----- - 2*x | dx
 |  |  ___       |   
 |  \\/ 5        /   
 |                   
/                    
0

$$\int\limits_{0}^{1} \left(- 2 x^{2} + \frac{x}{\sqrt{5}}\right)\, dx$$

Integral(x/sqrt(5) - 2*x^2, (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:
  
  $\int \left(- 2 x^{2}\right)\, dx = - 2 \int x^{2}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{2}\, dx = \frac{x^{3}}{3}$
  So, the result is: $- \frac{2 x^{3}}{3}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{x}{\sqrt{5}}\, dx = \frac{\sqrt{5}}{5} \int x\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x\, dx = \frac{x^{2}}{2}$
  So, the result is: $\frac{\frac{\sqrt{5}}{5} x^{2}}{2}$
The result is: $- \frac{2 x^{3}}{3} + \frac{\frac{\sqrt{5}}{5} x^{2}}{2}$
Now simplify:

$\frac{x^{2} \left(- 20 x + 3 \sqrt{5}\right)}{30}$
Add the constant of integration:

$\frac{x^{2} \left(- 20 x + 3 \sqrt{5}\right)}{30}+ \mathrm{constant}$

The answer is:

$\frac{x^{2} \left(- 20 x + 3 \sqrt{5}\right)}{30}+ \mathrm{constant}$

The answer (Indefinite) [src]

                                       ___
  /                                2 \/ 5 
 |                            3   x *-----
 | /  x        2\          2*x         5  
 | |----- - 2*x | dx = C - ---- + --------
 | |  ___       |           3        2    
 | \\/ 5        /                         
 |                                        
/

$$\int \left(- 2 x^{2} + \frac{x}{\sqrt{5}}\right)\, dx = C - \frac{2 x^{3}}{3} + \frac{\frac{\sqrt{5}}{5} x^{2}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

        ___
  2   \/ 5 
- - + -----
  3     10

$$- \frac{2}{3} + \frac{\sqrt{5}}{10}$$

        ___
  2   \/ 5 
- - + -----
  3     10

$$- \frac{2}{3} + \frac{\sqrt{5}}{10}$$

-2/3 + sqrt(5)/10

Numerical answer [src]

-0.443059868916688

-0.443059868916688

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

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Integral of xdx/sqrt5-2x^2 dx

Limits of integration:

The graph:

Piecewise:

The solution