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Similar expressions
1/4*((x-1)*(4x-1))
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Integral of 1/4((x+1)(4x-1)) dx

The solution

You have entered [src]

  3                     
  /                     
 |                      
 |  (x + 1)*(4*x - 1)   
 |  ----------------- dx
 |          4           
 |                      
/                       
1

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

Integral(((x + 1)*(4*x - 1))/4, (x, 1, 3))

Detail solution

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

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

$\frac{x \left(8 x^{2} + 9 x - 6\right)}{24}$
Add the constant of integration:

$\frac{x \left(8 x^{2} + 9 x - 6\right)}{24}+ \mathrm{constant}$

The answer is:

$\frac{x \left(8 x^{2} + 9 x - 6\right)}{24}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                        
 |                                 3      2
 | (x + 1)*(4*x - 1)          x   x    3*x 
 | ----------------- dx = C - - + -- + ----
 |         4                  4   3     8  
 |                                         
/

$$\int \frac{\left(x + 1\right) \left(4 x - 1\right)}{4}\, dx = C + \frac{x^{3}}{3} + \frac{3 x^{2}}{8} - \frac{x}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

67/6

$$\frac{67}{6}$$

67/6

$$\frac{67}{6}$$

67/6

Numerical answer [src]

11.1666666666667

11.1666666666667

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

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Identical expressions

Similar expressions

Integral of 1/4((x+1)(4x-1)) dx

Limits of integration:

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of 1/4*((x+1)*(4x-1)) dx

Limits of integration:

The graph:

Piecewise:

The solution

Integral of 1/4((x+1)(4x-1)) dx