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Integral of (4x^3-x^2-x^3)/2x dx

The solution

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  3                    
  /                    
 |                     
 |     3    2    3     
 |  4*x  - x  - x      
 |  --------------*x dx
 |        2            
 |                     
/                      
1

$$\int\limits_{1}^{3} x \frac{- x^{3} + \left(4 x^{3} - x^{2}\right)}{2}\, dx$$

Integral(((4*x^3 - x^2 - x^3)/2)*x, (x, 1, 3))

Detail solution

Rewrite the integrand:

$x \frac{- x^{3} + \left(4 x^{3} - x^{2}\right)}{2} = \frac{3 x^{4}}{2} - \frac{x^{3}}{2}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{3 x^{4}}{2}\, dx = \frac{3 \int x^{4}\, dx}{2}$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{4}\, dx = \frac{x^{5}}{5}$
  So, the result is: $\frac{3 x^{5}}{10}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{x^{3}}{2}\right)\, dx = - \frac{\int x^{3}\, dx}{2}$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{3}\, dx = \frac{x^{4}}{4}$
  So, the result is: $- \frac{x^{4}}{8}$
The result is: $\frac{3 x^{5}}{10} - \frac{x^{4}}{8}$
Now simplify:

$\frac{x^{4} \left(12 x - 5\right)}{40}$
Add the constant of integration:

$\frac{x^{4} \left(12 x - 5\right)}{40}+ \mathrm{constant}$

The answer is:

$\frac{x^{4} \left(12 x - 5\right)}{40}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                   
 |                                    
 |    3    2    3             4      5
 | 4*x  - x  - x             x    3*x 
 | --------------*x dx = C - -- + ----
 |       2                   8     10 
 |                                    
/

$$\int x \frac{- x^{3} + \left(4 x^{3} - x^{2}\right)}{2}\, dx = C + \frac{3 x^{5}}{10} - \frac{x^{4}}{8}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

313/5

$$\frac{313}{5}$$

313/5

$$\frac{313}{5}$$

313/5

Numerical answer [src]

62.6

62.6

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

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Integral of (4x^3-x^2-x^3)/2x dx

Limits of integration:

The graph:

Piecewise:

The solution