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Integral of (ln2*x)/(x*ln4*x) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1              
  /              
 |               
 |   log(2*x)    
 |  ---------- dx
 |  x*log(4*x)   
 |               
/                
0                
$$\int\limits_{0}^{1} \frac{\log{\left(2 x \right)}}{x \log{\left(4 x \right)}}\, dx$$
Integral(log(2*x)/((x*log(4*x))), (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

    2. Let .

      Then let and substitute :

      1. Rewrite the integrand:

      2. Integrate term-by-term:

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

        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 is .

            Now substitute back in:

          So, the result is:

        The result is:

      Now substitute back in:

    Method #2

    1. Use integration by parts:

      Let and let .

      Then .

      To find :

      1. Let .

        Then let and substitute :

        1. The integral of is .

        Now substitute back in:

      Now evaluate the sub-integral.

    2. 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. Let .

              Then let and substitute :

              1. Use integration by parts:

                Let and let .

                Then .

                To find :

                1. The integral of the exponential function is itself.

                Now evaluate the sub-integral.

              2. The integral of the exponential function is itself.

              Now substitute back in:

            So, the result is:

          Now substitute back in:

        So, the result is:

      Now substitute back in:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                          
 |                                                           
 |  log(2*x)                                                 
 | ---------- dx = C - log(2)*log(2*log(2) + log(x)) + log(x)
 | x*log(4*x)                                                
 |                                                           
/                                                            
$$\int \frac{\log{\left(2 x \right)}}{x \log{\left(4 x \right)}}\, dx = C + \log{\left(x \right)} - \log{\left(2 \right)} \log{\left(\log{\left(x \right)} + 2 \log{\left(2 \right)} \right)}$$
The graph
The answer [src]
nan
$$\text{NaN}$$
=
=
nan
$$\text{NaN}$$
nan
Numerical answer [src]
59.6786882263115
59.6786882263115

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