Integral of x-lnx dx

The solution

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

Integral(x - log(x), (x, 0, 1))

Detail solution

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

3/2

$$\frac{3}{2}$$

3/2

$$\frac{3}{2}$$

3/2

Numerical answer [src]

1.5

1.5

The graph

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

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Identical expressions

Similar expressions

Integral of x-lnx dx

Limits of integration:

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Piecewise:

The solution