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Integral of 2(lnx+1)/x dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  2                  
 e                   
  /                  
 |                   
 |  2*(log(x) + 1)   
 |  -------------- dx
 |        x          
 |                   
/                    
E                    
$$\int\limits_{e}^{e^{2}} \frac{2 \left(\log{\left(x \right)} + 1\right)}{x}\, dx$$
Integral((2*(log(x) + 1))/x, (x, E, exp(2)))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Let .

      Then let and substitute :

      1. Integrate term-by-term:

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

          1. The integral of is when :

          So, the result is:

        1. The integral of a constant is the constant times the variable of integration:

        The result is:

      Now substitute back in:

    Method #2

    1. Rewrite the integrand:

    2. Integrate term-by-term:

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

        1. Let .

          Then let and substitute :

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

            1. Let .

              Then let and substitute :

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

                1. The integral of is when :

                So, the result is:

              Now substitute back in:

            So, the result is:

          Now substitute back in:

        So, the result is:

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

        1. The integral of is .

        So, the result is:

      The result is:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                          
 |                                           
 | 2*(log(x) + 1)             2              
 | -------------- dx = C + log (x) + 2*log(x)
 |       x                                   
 |                                           
/                                            
$$\int \frac{2 \left(\log{\left(x \right)} + 1\right)}{x}\, dx = C + \log{\left(x \right)}^{2} + 2 \log{\left(x \right)}$$
The graph
The answer [src]
5
$$5$$
=
=
5
$$5$$
5
Numerical answer [src]
5.0
5.0

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