Mister Exam

Other calculators

Solving integrals/ xdx/sqrt5-2x^2

Integral of xdx/sqrt5-2x^2 dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1                  
  /                  
 |                   
 |  /  x        2\   
 |  |----- - 2*x | dx
 |  |  ___       |   
 |  \\/ 5        /   
 |                   
/                    
0
∫01(−2x2+x5) dx\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

  1. Integrate term-by-term:

    1. The integral of a constant times a function is the constant times the integral of the function:

      ∫(−2x2) dx=−2∫x2 dx\int \left(- 2 x^{2}\right)\, dx = - 2 \int x^{2}\, dx

      1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

        ∫x2 dx=x33\int x^{2}\, dx = \frac{x^{3}}{3}

      So, the result is: −2x33- \frac{2 x^{3}}{3}

    1. The integral of a constant times a function is the constant times the integral of the function:

      ∫x5 dx=55∫x dx\int \frac{x}{\sqrt{5}}\, dx = \frac{\sqrt{5}}{5} \int x\, dx

      1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

        ∫x dx=x22\int x\, dx = \frac{x^{2}}{2}

      So, the result is: 55x22\frac{\frac{\sqrt{5}}{5} x^{2}}{2}

    The result is: −2x33+55x22- \frac{2 x^{3}}{3} + \frac{\frac{\sqrt{5}}{5} x^{2}}{2}

  2. Now simplify:

    x2(−20x+35)30\frac{x^{2} \left(- 20 x + 3 \sqrt{5}\right)}{30}

  3. Add the constant of integration:

    x2(−20x+35)30+constant\frac{x^{2} \left(- 20 x + 3 \sqrt{5}\right)}{30}+ \mathrm{constant}

The answer is:

x2(−20x+35)30+constant\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        /                         
 |                                        
/
∫(−2x2+x5) dx=C−2x33+55x22\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
0.001.000.100.200.300.400.500.600.700.800.902-2
Plot the graph f(x)
The answer [src] 
        ___
  2   \/ 5 
- - + -----
  3     10
−23+510- \frac{2}{3} + \frac{\sqrt{5}}{10} 
=
= 
        ___
  2   \/ 5 
- - + -----
  3     10
−23+510- \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.

    © Mister Exam – Calculator