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Solving integrals/ 1/x(lnx)⁸

Integral of 1/x(lnx)⁸ dx

Limits of integration:

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The solution

You have entered [src] 
 oo           
  /           
 |            
 |     8      
 |  log (x)   
 |  ------- dx
 |     x      
 |            
/             
0
∫0∞log⁡(x)8x dx\int\limits_{0}^{\infty} \frac{\log{\left(x \right)}^{8}}{x}\, dx 
Integral(log(x)^8/x, (x, 0, oo))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

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

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

      ∫(−log⁡(1u)8u) du\int \left(- \frac{\log{\left(\frac{1}{u} \right)}^{8}}{u}\right)\, du

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

        ∫log⁡(1u)8u du=−∫log⁡(1u)8u du\int \frac{\log{\left(\frac{1}{u} \right)}^{8}}{u}\, du = - \int \frac{\log{\left(\frac{1}{u} \right)}^{8}}{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:

          ∫(−u8) du\int \left(- u^{8}\right)\, du

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

            ∫u8 du=−∫u8 du\int u^{8}\, du = - \int u^{8}\, du

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

              ∫u8 du=u99\int u^{8}\, du = \frac{u^{9}}{9}

            So, the result is: −u99- \frac{u^{9}}{9}

          Now substitute uu back in:

          −log⁡(1u)99- \frac{\log{\left(\frac{1}{u} \right)}^{9}}{9}

        So, the result is: log⁡(1u)99\frac{\log{\left(\frac{1}{u} \right)}^{9}}{9}

      Now substitute uu back in:

      log⁡(x)99\frac{\log{\left(x \right)}^{9}}{9}

    Method #2

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

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

      ∫u8 du\int u^{8}\, du

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

        ∫u8 du=u99\int u^{8}\, du = \frac{u^{9}}{9}

      Now substitute uu back in:

      log⁡(x)99\frac{\log{\left(x \right)}^{9}}{9}

  2. Add the constant of integration:

    log⁡(x)99+constant\frac{\log{\left(x \right)}^{9}}{9}+ \mathrm{constant}

The answer is:

log⁡(x)99+constant\frac{\log{\left(x \right)}^{9}}{9}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                        
 |                         
 |    8                9   
 | log (x)          log (x)
 | ------- dx = C + -------
 |    x                9   
 |                         
/
∫log⁡(x)8x dx=C+log⁡(x)99\int \frac{\log{\left(x \right)}^{8}}{x}\, dx = C + \frac{\log{\left(x \right)}^{9}}{9}
Simplify
The answer [src] 
oo
∞\infty 
=
= 
oo
∞\infty 
oo

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

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