Integral of yln(x) dy

The solution

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   ___           
 \/ x            
   /             
  |              
  |   y*log(x) dy
  |              
 /               
 1               
 -               
 x

$$\int\limits_{\frac{1}{x}}^{\sqrt{x}} y \log{\left(x \right)}\, dy$$

Integral(y*log(x), (y, 1/x, sqrt(x)))

Detail solution

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

$\int y \log{\left(x \right)}\, dy = \log{\left(x \right)} \int y\, dy$
1. The integral of $y^{n}$ is $\frac{y^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int y\, dy = \frac{y^{2}}{2}$
So, the result is: $\frac{y^{2} \log{\left(x \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                   2       
 |                   y *log(x)
 | y*log(x) dy = C + ---------
 |                       2    
/

$$\int y \log{\left(x \right)}\, dy = C + \frac{y^{2} \log{\left(x \right)}}{2}$$

Simplify

The answer [src]

x*log(x)   log(x)
-------- - ------
   2           2 
            2*x

$$\frac{x \log{\left(x \right)}}{2} - \frac{\log{\left(x \right)}}{2 x^{2}}$$

x*log(x)   log(x)
-------- - ------
   2           2 
            2*x

$$\frac{x \log{\left(x \right)}}{2} - \frac{\log{\left(x \right)}}{2 x^{2}}$$

x*log(x)/2 - log(x)/(2*x^2)

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

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Identical expressions

Integral of yln(x) dy

Limits of integration:

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The solution