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6x³+x²-2x+ one /2x- one
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Similar expressions
6x³+x²+2x+1/2x-1
6x³+x²-2x+1/2x+1
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6x³-x²-2x+1/2x-1

Solving integrals/ 6x³+x²-2x+1/2x-1

Integral of 6x³+x²-2x+1/2x-1 dx

The solution

You have entered [src]

  1                             
  /                             
 |                              
 |  /   3    2         x    \   
 |  |6*x  + x  - 2*x + - - 1| dx
 |  \                  2    /   
 |                              
/                               
0

$$\int\limits_{0}^{1} \left(\left(\frac{x}{2} + \left(- 2 x + \left(6 x^{3} + x^{2}\right)\right)\right) - 1\right)\, dx$$

Integral(6*x^3 + x^2 - 2*x + x/2 - 1, (x, 0, 1))

Detail solution

Integrate term-by-term:
1. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{x}{2}\, dx = \frac{\int x\, dx}{2}$
    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: $\frac{x^{2}}{4}$
  1. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 2 x\right)\, dx = - 2 \int x\, dx$
      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: $- x^{2}$
    1. Integrate term-by-term:
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int 6 x^{3}\, dx = 6 \int x^{3}\, dx$
        
        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: $\frac{3 x^{4}}{2}$
      1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
        
        $\int x^{2}\, dx = \frac{x^{3}}{3}$
      The result is: $\frac{3 x^{4}}{2} + \frac{x^{3}}{3}$
    The result is: $\frac{3 x^{4}}{2} + \frac{x^{3}}{3} - x^{2}$
  The result is: $\frac{3 x^{4}}{2} + \frac{x^{3}}{3} - \frac{3 x^{2}}{4}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int \left(-1\right)\, dx = - x$
The result is: $\frac{3 x^{4}}{2} + \frac{x^{3}}{3} - \frac{3 x^{2}}{4} - x$
Now simplify:

$\frac{x \left(18 x^{3} + 4 x^{2} - 9 x - 12\right)}{12}$
Add the constant of integration:

$\frac{x \left(18 x^{3} + 4 x^{2} - 9 x - 12\right)}{12}+ \mathrm{constant}$

The answer is:

$\frac{x \left(18 x^{3} + 4 x^{2} - 9 x - 12\right)}{12}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                       
 |                                           2    3      4
 | /   3    2         x    \              3*x    x    3*x 
 | |6*x  + x  - 2*x + - - 1| dx = C - x - ---- + -- + ----
 | \                  2    /               4     3     2  
 |                                                        
/

$$\int \left(\left(\frac{x}{2} + \left(- 2 x + \left(6 x^{3} + x^{2}\right)\right)\right) - 1\right)\, dx = C + \frac{3 x^{4}}{2} + \frac{x^{3}}{3} - \frac{3 x^{2}}{4} - x$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1/12

$$\frac{1}{12}$$

1/12

$$\frac{1}{12}$$

1/12

Numerical answer [src]

0.0833333333333333

0.0833333333333333

The graph

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

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Integral of 6x³+x²-2x+1/2x-1 dx

Limits of integration:

The graph:

Piecewise:

The solution