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Solving integrals/ 2x/(ln(x))

Integral of 2x/(ln(x)) dx

Limits of integration:

from to
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Piecewise:

The solution

You have entered [src] 
  n          
  /          
 |           
 |   2*x     
 |  ------ dx
 |  log(x)   
 |           
/            
0
∫0n2xlog⁡(x) dx\int\limits_{0}^{n} \frac{2 x}{\log{\left(x \right)}}\, dx 
Integral((2*x)/log(x), (x, 0, n))
Detail solution

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

    Then let du=dxxdu = \frac{dx}{x} and substitute 2du2 du:

    ∫2e2uu du\int \frac{2 e^{2 u}}{u}\, du

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

      ∫e2uu du=2∫e2uu du\int \frac{e^{2 u}}{u}\, du = 2 \int \frac{e^{2 u}}{u}\, du

        EiRule(a=2, b=0, context=exp(2*_u)/_u, symbol=_u)

      So, the result is: 2Ei⁡(2u)2 \operatorname{Ei}{\left(2 u \right)}

    Now substitute uu back in:

    2Ei⁡(2log⁡(x))2 \operatorname{Ei}{\left(2 \log{\left(x \right)} \right)}

  2. Add the constant of integration:

    2Ei⁡(2log⁡(x))+constant2 \operatorname{Ei}{\left(2 \log{\left(x \right)} \right)}+ \mathrm{constant}

The answer is:

2Ei⁡(2log⁡(x))+constant2 \operatorname{Ei}{\left(2 \log{\left(x \right)} \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                              
 |                               
 |  2*x                          
 | ------ dx = C + 2*Ei(2*log(x))
 | log(x)                        
 |                               
/
∫2xlog⁡(x) dx=C+2Ei⁡(2log⁡(x))\int \frac{2 x}{\log{\left(x \right)}}\, dx = C + 2 \operatorname{Ei}{\left(2 \log{\left(x \right)} \right)}
Simplify
The answer [src] 
2*Ei(2*log(n))
2Ei⁡(2log⁡(n))2 \operatorname{Ei}{\left(2 \log{\left(n \right)} \right)} 
=
= 
2*Ei(2*log(n))
2Ei⁡(2log⁡(n))2 \operatorname{Ei}{\left(2 \log{\left(n \right)} \right)} 
2*Ei(2*log(n))

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

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