Mister Exam

Other calculators

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

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1             
  /             
 |              
 |     2        
 |  log (2*x)   
 |  --------- dx
 |      x       
 |              
/               
0
∫01log⁡(2x)2x dx\int\limits_{0}^{1} \frac{\log{\left(2 x \right)}^{2}}{x}\, dx 
Integral(log(2*x)^2/x, (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=log⁡(2x)u = \log{\left(2 x \right)}.

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

      ∫u2 du\int u^{2}\, du

      1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

        ∫u2 du=u33\int u^{2}\, du = \frac{u^{3}}{3}

      Now substitute uu back in:

      log⁡(2x)33\frac{\log{\left(2 x \right)}^{3}}{3}

    Method #2

    1. Rewrite the integrand:

      log⁡(2x)2x=log⁡(x)2+2log⁡(2)log⁡(x)+log⁡(2)2x\frac{\log{\left(2 x \right)}^{2}}{x} = \frac{\log{\left(x \right)}^{2} + 2 \log{\left(2 \right)} \log{\left(x \right)} + \log{\left(2 \right)}^{2}}{x}

    2. Let u=1xu = \frac{1}{x}.

      Then let du=−dxx2du = - \frac{dx}{x^{2}} and substitute dudu:

      ∫−log⁡(1u)2−2log⁡(2)log⁡(1u)−log⁡(2)2u du\int \frac{- \log{\left(\frac{1}{u} \right)}^{2} - 2 \log{\left(2 \right)} \log{\left(\frac{1}{u} \right)} - \log{\left(2 \right)}^{2}}{u}\, du

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

        Then let du=−duudu = - \frac{du}{u} and substitute dudu:

        ∫(u2+2ulog⁡(2)+log⁡(2)2) du\int \left(u^{2} + 2 u \log{\left(2 \right)} + \log{\left(2 \right)}^{2}\right)\, du

        1. Integrate term-by-term:

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u2 du=u33\int u^{2}\, du = \frac{u^{3}}{3}

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

            ∫2ulog⁡(2) du=2log⁡(2)∫u du\int 2 u \log{\left(2 \right)}\, du = 2 \log{\left(2 \right)} \int u\, du

            1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

              ∫u du=u22\int u\, du = \frac{u^{2}}{2}

            So, the result is: u2log⁡(2)u^{2} \log{\left(2 \right)}

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

            ∫log⁡(2)2 du=ulog⁡(2)2\int \log{\left(2 \right)}^{2}\, du = u \log{\left(2 \right)}^{2}

          The result is: u33+u2log⁡(2)+ulog⁡(2)2\frac{u^{3}}{3} + u^{2} \log{\left(2 \right)} + u \log{\left(2 \right)}^{2}

        Now substitute uu back in:

        log⁡(1u)33+log⁡(2)log⁡(1u)2+log⁡(2)2log⁡(1u)\frac{\log{\left(\frac{1}{u} \right)}^{3}}{3} + \log{\left(2 \right)} \log{\left(\frac{1}{u} \right)}^{2} + \log{\left(2 \right)}^{2} \log{\left(\frac{1}{u} \right)}

      Now substitute uu back in:

      log⁡(x)33+log⁡(2)log⁡(x)2+log⁡(2)2log⁡(x)\frac{\log{\left(x \right)}^{3}}{3} + \log{\left(2 \right)} \log{\left(x \right)}^{2} + \log{\left(2 \right)}^{2} \log{\left(x \right)}

    Method #3

    1. Rewrite the integrand:

      log⁡(2x)2x=log⁡(x)2x+2log⁡(2)log⁡(x)x+log⁡(2)2x\frac{\log{\left(2 x \right)}^{2}}{x} = \frac{\log{\left(x \right)}^{2}}{x} + \frac{2 \log{\left(2 \right)} \log{\left(x \right)}}{x} + \frac{\log{\left(2 \right)}^{2}}{x}

    2. Integrate term-by-term:

      1. Let u=1xu = \frac{1}{x}.

        Then let du=−dxx2du = - \frac{dx}{x^{2}} and substitute −du- du:

        ∫log⁡(1u)2u du\int \frac{\log{\left(\frac{1}{u} \right)}^{2}}{u}\, du

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

          ∫(−log⁡(1u)2u) du=−∫log⁡(1u)2u du\int \left(- \frac{\log{\left(\frac{1}{u} \right)}^{2}}{u}\right)\, du = - \int \frac{\log{\left(\frac{1}{u} \right)}^{2}}{u}\, du

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

            Then let du=−duudu = - \frac{du}{u} and substitute −du- du:

            ∫u2 du\int u^{2}\, du

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

              ∫(−u2) du=−∫u2 du\int \left(- u^{2}\right)\, du = - \int u^{2}\, du

              1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

                ∫u2 du=u33\int u^{2}\, du = \frac{u^{3}}{3}

              So, the result is: −u33- \frac{u^{3}}{3}

            Now substitute uu back in:

            −log⁡(1u)33- \frac{\log{\left(\frac{1}{u} \right)}^{3}}{3}

          So, the result is: log⁡(1u)33\frac{\log{\left(\frac{1}{u} \right)}^{3}}{3}

        Now substitute uu back in:

        log⁡(x)33\frac{\log{\left(x \right)}^{3}}{3}

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

        ∫2log⁡(2)log⁡(x)x dx=2log⁡(2)∫log⁡(x)x dx\int \frac{2 \log{\left(2 \right)} \log{\left(x \right)}}{x}\, dx = 2 \log{\left(2 \right)} \int \frac{\log{\left(x \right)}}{x}\, dx

        1. Let u=1xu = \frac{1}{x}.

          Then let du=−dxx2du = - \frac{dx}{x^{2}} and substitute −du- du:

          ∫log⁡(1u)u du\int \frac{\log{\left(\frac{1}{u} \right)}}{u}\, du

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

            ∫(−log⁡(1u)u) du=−∫log⁡(1u)u du\int \left(- \frac{\log{\left(\frac{1}{u} \right)}}{u}\right)\, du = - \int \frac{\log{\left(\frac{1}{u} \right)}}{u}\, du

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

              Then let du=−duudu = - \frac{du}{u} and substitute −du- du:

              ∫u du\int u\, du

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

                ∫(−u) du=−∫u du\int \left(- u\right)\, du = - \int u\, du

                1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

                  ∫u du=u22\int u\, du = \frac{u^{2}}{2}

                So, the result is: −u22- \frac{u^{2}}{2}

              Now substitute uu back in:

              −log⁡(1u)22- \frac{\log{\left(\frac{1}{u} \right)}^{2}}{2}

            So, the result is: log⁡(1u)22\frac{\log{\left(\frac{1}{u} \right)}^{2}}{2}

          Now substitute uu back in:

          log⁡(x)22\frac{\log{\left(x \right)}^{2}}{2}

        So, the result is: log⁡(2)log⁡(x)2\log{\left(2 \right)} \log{\left(x \right)}^{2}

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

        ∫log⁡(2)2x dx=log⁡(2)2∫1x dx\int \frac{\log{\left(2 \right)}^{2}}{x}\, dx = \log{\left(2 \right)}^{2} \int \frac{1}{x}\, dx

        1. The integral of 1x\frac{1}{x} is log⁡(x)\log{\left(x \right)}.

        So, the result is: log⁡(2)2log⁡(x)\log{\left(2 \right)}^{2} \log{\left(x \right)}

      The result is: log⁡(x)33+log⁡(2)log⁡(x)2+log⁡(2)2log⁡(x)\frac{\log{\left(x \right)}^{3}}{3} + \log{\left(2 \right)} \log{\left(x \right)}^{2} + \log{\left(2 \right)}^{2} \log{\left(x \right)}

  2. Add the constant of integration:

    log⁡(2x)33+constant\frac{\log{\left(2 x \right)}^{3}}{3}+ \mathrm{constant}

The answer is:

log⁡(2x)33+constant\frac{\log{\left(2 x \right)}^{3}}{3}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                            
 |                             
 |    2                  3     
 | log (2*x)          log (2*x)
 | --------- dx = C + ---------
 |     x                  3    
 |                             
/
(log⁡(2 x))33{{\left(\log \left(2\,x\right)\right)^3}\over{3}}
Simplify
The answer [src] 
oo
%a{\it \%a} 
=
= 
oo
∞\infty
Упростить
Numerical answer [src] 
27242.1350802983
27242.1350802983

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

    © Mister Exam – Calculator