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Solving integrals/ x^3+logx-2x

Integral of x^3+logx-2x dx

The solution

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  1                       
  /                       
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 |  / 3               \   
 |  \x  + log(x) - 2*x/ dx
 |                        
/                         
0

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

Integral(x^3 + log(x) - 2*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:
  
  $\int \left(- 2 x\right)\, dx = - 2 \int x\, dx$
  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: $- x^{2}$
1. Integrate term-by-term:
  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. Use integration by parts:
    
    $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
    
    Let $u{\left(x \right)} = \log{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = 1$ .
    
    Then $\operatorname{du}{\left(x \right)} = \frac{1}{x}$ .
    
    To find $v{\left(x \right)}$ :
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, dx = x$
    Now evaluate the sub-integral.
  2. The integral of a constant is the constant times the variable of integration:
    
    $\int 1\, dx = x$
  The result is: $\frac{x^{4}}{4} + x \log{\left(x \right)} - x$
The result is: $\frac{x^{4}}{4} - x^{2} + x \log{\left(x \right)} - x$
Now simplify:

$x \left(\frac{x^{3}}{4} - x + \log{\left(x \right)} - 1\right)$
Add the constant of integration:

$x \left(\frac{x^{3}}{4} - x + \log{\left(x \right)} - 1\right)+ \mathrm{constant}$

The answer is:

$x \left(\frac{x^{3}}{4} - x + \log{\left(x \right)} - 1\right)+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                   
 |                                        4           
 | / 3               \               2   x            
 | \x  + log(x) - 2*x/ dx = C - x - x  + -- + x*log(x)
 |                                       4            
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

-7/4

$$- \frac{7}{4}$$

-7/4

$$- \frac{7}{4}$$

-7/4

Numerical answer [src]

-1.75

-1.75

The graph

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

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Integral of x^3+logx-2x dx

Limits of integration:

The graph:

Piecewise:

The solution