Integral of 12+x-x²+x³+x⁴ dx

The solution

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  1                           
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 |  /          2    3    4\   
 |  \12 + x - x  + x  + x / dx
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$$\int\limits_{0}^{1} \left(x^{4} + x^{3} - x^{2} + x + 12\right)\, dx$$

Integral(12 + x - x^2 + x^3 + x^4, (x, 0, 1))

Detail solution

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 $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x^{3}\, dx = \frac{x^{4}}{4}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- x^{2}\right)\, dx = - \int x^{2}\, dx$
  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: $- \frac{x^{3}}{3}$
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x\, dx = \frac{x^{2}}{2}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 12\, dx = 12 x$
The result is: $\frac{x^{5}}{5} + \frac{x^{4}}{4} - \frac{x^{3}}{3} + \frac{x^{2}}{2} + 12 x$
Now simplify:

$\frac{x \left(12 x^{4} + 15 x^{3} - 20 x^{2} + 30 x + 720\right)}{60}$
Add the constant of integration:

$\frac{x \left(12 x^{4} + 15 x^{3} - 20 x^{2} + 30 x + 720\right)}{60}+ \mathrm{constant}$

The answer is:

$\frac{x \left(12 x^{4} + 15 x^{3} - 20 x^{2} + 30 x + 720\right)}{60}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                         
 |                                   2           3    4    5
 | /          2    3    4\          x           x    x    x 
 | \12 + x - x  + x  + x / dx = C + -- + 12*x - -- + -- + --
 |                                  2           3    4    5 
/

$${{x^5}\over{5}}+{{x^4}\over{4}}-{{x^3}\over{3}}+{{x^2}\over{2}}+12 \,x$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

757
---
 60

$${{757}\over{60}}$$

757
---
 60

$$\frac{757}{60}$$

Simplify

Numerical answer [src]

12.6166666666667

12.6166666666667

The graph

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

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Integral of 12+x-x²+x³+x⁴ dx

Limits of integration:

The graph:

Piecewise:

The solution