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Solving integrals/ (sqrt(1-lnx))/x

Integral of (sqrt(1-lnx))/x dx

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The solution

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∫011−log⁡(x)x dx\int\limits_{0}^{1} \frac{\sqrt{1 - \log{\left(x \right)}}}{x}\, dx 
Integral(sqrt(1 - log(x))/x, (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

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

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

      ∫(−u) du\int \left(- \sqrt{u}\right)\, du

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

        ∫u du=−∫u du\int \sqrt{u}\, du = - \int \sqrt{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=2u323\int \sqrt{u}\, du = \frac{2 u^{\frac{3}{2}}}{3}

        So, the result is: −2u323- \frac{2 u^{\frac{3}{2}}}{3}

      Now substitute uu back in:

      −2(1−log⁡(x))323- \frac{2 \left(1 - \log{\left(x \right)}\right)^{\frac{3}{2}}}{3}

    Method #2

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

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

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

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

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

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

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

          ∫u du\int \sqrt{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=2u323\int \sqrt{u}\, du = \frac{2 u^{\frac{3}{2}}}{3}

          Now substitute uu back in:

          2(1−log⁡(1u))323\frac{2 \left(1 - \log{\left(\frac{1}{u} \right)}\right)^{\frac{3}{2}}}{3}

        So, the result is: −2(1−log⁡(1u))323- \frac{2 \left(1 - \log{\left(\frac{1}{u} \right)}\right)^{\frac{3}{2}}}{3}

      Now substitute uu back in:

      −2(1−log⁡(x))323- \frac{2 \left(1 - \log{\left(x \right)}\right)^{\frac{3}{2}}}{3}

  2. Add the constant of integration:

    −2(1−log⁡(x))323+constant- \frac{2 \left(1 - \log{\left(x \right)}\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}

The answer is:

−2(1−log⁡(x))323+constant- \frac{2 \left(1 - \log{\left(x \right)}\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                         
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 | \/ 1 - log(x)           2*(1 - log(x))   
 | -------------- dx = C - -----------------
 |       x                         3        
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∫1−log⁡(x)x dx=C−2(1−log⁡(x))323\int \frac{\sqrt{1 - \log{\left(x \right)}}}{x}\, dx = C - \frac{2 \left(1 - \log{\left(x \right)}\right)^{\frac{3}{2}}}{3}
Simplify
The answer [src] 
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Numerical answer [src] 
201.185008592488
201.185008592488

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

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