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Solving integrals/ ∫(lnx)²/xdx

You entered:

∫(lnx)²/xdx

What you mean?

log(x)^2*dx/x
 
log(x)^(2/x)*dx
 
log(x)^(2/x)*dx

Integral of ∫(lnx)²/xdx dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
  1               
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 |     2    1     
 |  log (x)*-*1 dx
 |          x     
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0
∫01log⁡(x)2⋅1x1 dx\int\limits_{0}^{1} \log{\left(x \right)}^{2} \cdot \frac{1}{x} 1\, dx 
Integral(log(x)^2*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:

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

    Method #2

    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}

  2. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                            
 |                         3   
 |    2    1            log (x)
 | log (x)*-*1 dx = C + -------
 |         x               3   
 |                             
/
(log⁡x)33{{\left(\log x\right)^3}\over{3}}
Simplify
The answer [src] 
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∞\infty
Simplify
Numerical answer [src] 
28568.3797156332
28568.3797156332

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

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