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. 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: $\frac{x^{2}}{2} + x \log{\left(x \right)} - x$
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]

  /                       2               
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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]

-1/2

$$- \frac{1}{2}$$

-1/2

$$- \frac{1}{2}$$

-1/2

Numerical answer [src]

-0.5

-0.5

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:

The graph:

Piecewise:

The solution