Integral of S(2x⁴+8x³-3x²+x)dx dx

The solution

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 |  s*\2*x  + 8*x  - 3*x  + x/*1 dx
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$$\int\limits_{0}^{1} s \left(2 x^{4} + 8 x^{3} - 3 x^{2} + x\right) 1\, dx$$

Integral(s*(2*x^4 + 8*x^3 - 3*x^2 + x)*1, (x, 0, 1))

Detail solution

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

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

$\frac{s x^{2} \cdot \left(4 x^{3} + 20 x^{2} - 10 x + 5\right)}{10}$
Add the constant of integration:

$\frac{s x^{2} \cdot \left(4 x^{3} + 20 x^{2} - 10 x + 5\right)}{10}+ \mathrm{constant}$

The answer is:

$\frac{s x^{2} \cdot \left(4 x^{3} + 20 x^{2} - 10 x + 5\right)}{10}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                               
 |                                         / 2                  5\
 |   /   4      3      2    \              |x     3      4   2*x |
 | s*\2*x  + 8*x  - 3*x  + x/*1 dx = C + s*|-- - x  + 2*x  + ----|
 |                                         \2                 5  /
/

$$\int s \left(2 x^{4} + 8 x^{3} - 3 x^{2} + x\right) 1\, dx = C + s \left(\frac{2 x^{5}}{5} + 2 x^{4} - x^{3} + \frac{x^{2}}{2}\right)$$

Simplify

The answer [src]

19*s
----
 10

$$\frac{19 s}{10}$$

19*s
----
 10

$$\frac{19 s}{10}$$

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

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Integral of S(2x⁴+8x³-3x²+x)dx dx

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