Integral of 2x^2-5x dx

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         ____               
   5   \/ 73                
 - - + ------               
   4     4                  
       /                    
      |                     
      |      /   2      \   
      |      \2*x  - 5*x/ dx
      |                     
     /                      
        ____                
  5   \/ 73                 
- - - ------                
  4     4

$$\int\limits_{- \frac{\sqrt{73}}{4} - \frac{5}{4}}^{- \frac{5}{4} + \frac{\sqrt{73}}{4}} \left(2 x^{2} - 5 x\right)\, dx$$

Integral(2*x^2 - 5*x, (x, -5/4 - sqrt(73)/4, -5/4 + sqrt(73)/4))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 2 x^{2}\, dx = 2 \int x^{2}\, dx$
  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:
  
  $\int \left(- 5 x\right)\, dx = - 5 \int x\, dx$
  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: $- \frac{5 x^{2}}{2}$
The result is: $\frac{2 x^{3}}{3} - \frac{5 x^{2}}{2}$
Now simplify:

$\frac{x^{2} \left(4 x - 15\right)}{6}$
Add the constant of integration:

$\frac{x^{2} \left(4 x - 15\right)}{6}+ \mathrm{constant}$

The answer is:

$\frac{x^{2} \left(4 x - 15\right)}{6}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

                  2                   3                   3                   2
    /        ____\      /        ____\      /        ____\      /        ____\ 
    |  5   \/ 73 |      |  5   \/ 73 |      |  5   \/ 73 |      |  5   \/ 73 | 
  5*|- - + ------|    2*|- - - ------|    2*|- - + ------|    5*|- - - ------| 
    \  4     4   /      \  4     4   /      \  4     4   /      \  4     4   / 
- ----------------- - ----------------- + ----------------- + -----------------
          2                   3                   3                   2

$$- \frac{5 \left(- \frac{5}{4} + \frac{\sqrt{73}}{4}\right)^{2}}{2} + \frac{2 \left(- \frac{5}{4} + \frac{\sqrt{73}}{4}\right)^{3}}{3} - \frac{2 \left(- \frac{\sqrt{73}}{4} - \frac{5}{4}\right)^{3}}{3} + \frac{5 \left(- \frac{\sqrt{73}}{4} - \frac{5}{4}\right)^{2}}{2}$$

=

                  2                   3                   3                   2
    /        ____\      /        ____\      /        ____\      /        ____\ 
    |  5   \/ 73 |      |  5   \/ 73 |      |  5   \/ 73 |      |  5   \/ 73 | 
  5*|- - + ------|    2*|- - - ------|    2*|- - + ------|    5*|- - - ------| 
    \  4     4   /      \  4     4   /      \  4     4   /      \  4     4   / 
- ----------------- - ----------------- + ----------------- + -----------------
          2                   3                   3                   2

$$- \frac{5 \left(- \frac{5}{4} + \frac{\sqrt{73}}{4}\right)^{2}}{2} + \frac{2 \left(- \frac{5}{4} + \frac{\sqrt{73}}{4}\right)^{3}}{3} - \frac{2 \left(- \frac{\sqrt{73}}{4} - \frac{5}{4}\right)^{3}}{3} + \frac{5 \left(- \frac{\sqrt{73}}{4} - \frac{5}{4}\right)^{2}}{2}$$

-5*(-5/4 + sqrt(73)/4)^2/2 - 2*(-5/4 - sqrt(73)/4)^3/3 + 2*(-5/4 + sqrt(73)/4)^3/3 + 5*(-5/4 - sqrt(73)/4)^2/2

Numerical answer [src]

53.0440232521797

53.0440232521797

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Integral of 2x^2-5x dx

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