Integral of (2x-1)*2x dx

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

Integral(((2*x - 1)*2)*x, (x, 0, 1))

Detail solution

Rewrite the integrand:

$x 2 \left(2 x - 1\right) = 4 x^{2} - 2 x$
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:
  
  $\int \left(- 2 x\right)\, dx = - 2 \int x\, dx$
  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: $- x^{2}$
The result is: $\frac{4 x^{3}}{3} - x^{2}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

1/3

$$\frac{1}{3}$$

1/3

$$\frac{1}{3}$$

1/3

Numerical answer [src]

0.333333333333333

0.333333333333333

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