Integral of sqrt(ln(x)) dx

The solution

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  1              
  /              
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 |    ________   
 |  \/ log(x)  dx
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/                
0

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

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

Detail solution

Let $u = \log{\left(x \right)}$ .

Then let $du = \frac{dx}{x}$ and substitute $du$ :

$\int \sqrt{u} e^{u}\, du$
Now substitute $u$ back in:

$\frac{\left(x \sqrt{- \log{\left(x \right)}} + \frac{\sqrt{\pi} \operatorname{erfc}{\left(\sqrt{- \log{\left(x \right)}} \right)}}{2}\right) \sqrt{\log{\left(x \right)}}}{\sqrt{- \log{\left(x \right)}}}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

                                  /                  ____     /  _________\\
  /                      ________ |    _________   \/ pi *erfc\\/ -log(x) /|
 |                     \/ log(x) *|x*\/ -log(x)  + ------------------------|
 |   ________                     \                           2            /
 | \/ log(x)  dx = C + -----------------------------------------------------
 |                                            _________                     
/                                           \/ -log(x)

$$\int \sqrt{\log{\left(x \right)}}\, dx = C + \frac{\left(x \sqrt{- \log{\left(x \right)}} + \frac{\sqrt{\pi} \operatorname{erfc}{\left(\sqrt{- \log{\left(x \right)}} \right)}}{2}\right) \sqrt{\log{\left(x \right)}}}{\sqrt{- \log{\left(x \right)}}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

    ____
I*\/ pi 
--------
   2

$$\frac{i \sqrt{\pi}}{2}$$

    ____
I*\/ pi 
--------
   2

$$\frac{i \sqrt{\pi}}{2}$$

i*sqrt(pi)/2

Numerical answer [src]

(0.0 + 0.886226925452758j)

(0.0 + 0.886226925452758j)

The graph

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

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Integral of sqrt(ln(x)) dx

Limits of integration:

The graph:

Piecewise:

The solution