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Solving integrals/ ln^3xdx/x

Integral of ln^3xdx/x dx

Limits of integration:

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The solution

You have entered [src] 
  1               
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∫01log⁡(x)3⋅1⋅1x dx\int\limits_{0}^{1} \log{\left(x \right)}^{3} \cdot 1 \cdot \frac{1}{x}\, dx 
Integral(log(x)^3*1/x, (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

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

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

      ∫u3 du\int u^{3}\, du

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

        ∫u3 du=u44\int u^{3}\, du = \frac{u^{4}}{4}

      Now substitute uu back in:

      log⁡(x)44\frac{\log{\left(x \right)}^{4}}{4}

    Method #2

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

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

      ∫log⁡(1u)3u du\int \frac{\log{\left(\frac{1}{u} \right)}^{3}}{u}\, du

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

        ∫(−log⁡(1u)3u) du=−∫log⁡(1u)3u du\int \left(- \frac{\log{\left(\frac{1}{u} \right)}^{3}}{u}\right)\, du = - \int \frac{\log{\left(\frac{1}{u} \right)}^{3}}{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:

          ∫u3 du\int u^{3}\, du

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

            ∫(−u3) du=−∫u3 du\int \left(- u^{3}\right)\, du = - \int u^{3}\, du

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

              ∫u3 du=u44\int u^{3}\, du = \frac{u^{4}}{4}

            So, the result is: −u44- \frac{u^{4}}{4}

          Now substitute uu back in:

          −log⁡(1u)44- \frac{\log{\left(\frac{1}{u} \right)}^{4}}{4}

        So, the result is: log⁡(1u)44\frac{\log{\left(\frac{1}{u} \right)}^{4}}{4}

      Now substitute uu back in:

      log⁡(x)44\frac{\log{\left(x \right)}^{4}}{4}

  2. Add the constant of integration:

    log⁡(x)44+constant\frac{\log{\left(x \right)}^{4}}{4}+ \mathrm{constant}

The answer is:

log⁡(x)44+constant\frac{\log{\left(x \right)}^{4}}{4}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                            
 |                         4   
 |    3      1          log (x)
 | log (x)*1*- dx = C + -------
 |           x             4   
 |                             
/
(log⁡x)44{{\left(\log x\right)^4}\over{4}}
Simplify
The answer [src] 
-oo
%a{\it \%a} 
=
= 
-oo
−∞-\infty
Simplify
Numerical answer [src] 
-944636.568261642
-944636.568261642

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

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