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

The solution

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  n          
  /          
 |           
 |   2*x     
 |  ------ dx
 |  log(x)   
 |           
/            
0

$$\int\limits_{0}^{n} \frac{2 x}{\log{\left(x \right)}}\, dx$$

Integral((2*x)/log(x), (x, 0, n))

Detail solution

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

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

$\int \frac{2 e^{2 u}}{u}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{e^{2 u}}{u}\, du = 2 \int \frac{e^{2 u}}{u}\, du$
  So, the result is: $2 \operatorname{Ei}{\left(2 u \right)}$
Now substitute $u$ back in:

$2 \operatorname{Ei}{\left(2 \log{\left(x \right)} \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                              
 |                               
 |  2*x                          
 | ------ dx = C + 2*Ei(2*log(x))
 | log(x)                        
 |                               
/

$$\int \frac{2 x}{\log{\left(x \right)}}\, dx = C + 2 \operatorname{Ei}{\left(2 \log{\left(x \right)} \right)}$$

Simplify

The answer [src]

2*Ei(2*log(n))

$$2 \operatorname{Ei}{\left(2 \log{\left(n \right)} \right)}$$

2*Ei(2*log(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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Integral of 2x/(ln(x)) dx

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The solution