Integral of (x⁴-8x³+4x) dx

The solution

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$$\int\limits_{0}^{1} \left(4 x + \left(x^{4} - 8 x^{3}\right)\right)\, dx$$

Integral(x^4 - 8*x^3 + 4*x, (x, 0, 1))

Detail solution

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

$\frac{x^{2} \left(x^{3} - 10 x^{2} + 10\right)}{5}$
Add the constant of integration:

$\frac{x^{2} \left(x^{3} - 10 x^{2} + 10\right)}{5}+ \mathrm{constant}$

The answer is:

$\frac{x^{2} \left(x^{3} - 10 x^{2} + 10\right)}{5}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

1/5

$$\frac{1}{5}$$

1/5

$$\frac{1}{5}$$

1/5

Numerical answer [src]

0.2

0.2

The graph

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

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Identical expressions

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Integral of (x⁴-8x³+4x) dx

Limits of integration:

The graph:

Piecewise:

The solution