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Solving integrals/ yln(x)

Integral of yln(x) dy

Limits of integration:

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Piecewise:

The solution

You have entered [src] 
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∫1xxylog⁡(x) dy\int\limits_{\frac{1}{x}}^{\sqrt{x}} y \log{\left(x \right)}\, dy 
Integral(y*log(x), (y, 1/x, sqrt(x)))
Detail solution

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

    ∫ylog⁡(x) dy=log⁡(x)∫y dy\int y \log{\left(x \right)}\, dy = \log{\left(x \right)} \int y\, dy

    1. The integral of yny^{n} is yn+1n+1\frac{y^{n + 1}}{n + 1} when n≠−1n \neq -1:

      ∫y dy=y22\int y\, dy = \frac{y^{2}}{2}

    So, the result is: y2log⁡(x)2\frac{y^{2} \log{\left(x \right)}}{2}

  2. Add the constant of integration:

    y2log⁡(x)2+constant\frac{y^{2} \log{\left(x \right)}}{2}+ \mathrm{constant}

The answer is:

y2log⁡(x)2+constant\frac{y^{2} \log{\left(x \right)}}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                   2       
 |                   y *log(x)
 | y*log(x) dy = C + ---------
 |                       2    
/
∫ylog⁡(x) dy=C+y2log⁡(x)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
xlog⁡(x)2−log⁡(x)2x2\frac{x \log{\left(x \right)}}{2} - \frac{\log{\left(x \right)}}{2 x^{2}} 
=
= 
x*log(x)   log(x)
-------- - ------
   2           2 
            2*x
xlog⁡(x)2−log⁡(x)2x2\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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