Integral of x+2lnx dx

The solution

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

Integral(x + 2*log(x), (x, -1, 1))

Detail solution

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

-4 + 2*pi*I

$$-4 + 2 i \pi$$

-4 + 2*pi*I

$$-4 + 2 i \pi$$

-4 + 2*pi*i

Numerical answer [src]

(-inf + 6.20607902279608j)

(-inf + 6.20607902279608j)

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

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Integral of x+2lnx dx

Limits of integration:

The graph:

Piecewise:

The solution