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Solving integrals/ 1/(x^(1/2)-1)

Integral of 1/(x^(1/2)-1) dx

Limits of integration:

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Piecewise:

The solution

You have entered [src] 
 oo             
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 |  \/ x  - 1   
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2
∫2∞1x−1 dx\int\limits_{2}^{\infty} \frac{1}{\sqrt{x} - 1}\, dx 
Integral(1/(sqrt(x) - 1), (x, 2, oo))
Detail solution

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

    Then let du=dx2xdu = \frac{dx}{2 \sqrt{x}} and substitute 2du2 du:

    ∫2uu−1 du\int \frac{2 u}{u - 1}\, du

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

      ∫uu−1 du=2∫uu−1 du\int \frac{u}{u - 1}\, du = 2 \int \frac{u}{u - 1}\, du

      1. Rewrite the integrand:

        uu−1=1+1u−1\frac{u}{u - 1} = 1 + \frac{1}{u - 1}

      2. Integrate term-by-term:

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

          ∫1 du=u\int 1\, du = u

        1. Let u=u−1u = u - 1.

          Then let du=dudu = du and substitute dudu:

          ∫1u du\int \frac{1}{u}\, du

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

          Now substitute uu back in:

          log⁡(u−1)\log{\left(u - 1 \right)}

        The result is: u+log⁡(u−1)u + \log{\left(u - 1 \right)}

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

    Now substitute uu back in:

    2x+2log⁡(x−1)2 \sqrt{x} + 2 \log{\left(\sqrt{x} - 1 \right)}

  2. Add the constant of integration:

    2x+2log⁡(x−1)+constant2 \sqrt{x} + 2 \log{\left(\sqrt{x} - 1 \right)}+ \mathrm{constant}

The answer is:

2x+2log⁡(x−1)+constant2 \sqrt{x} + 2 \log{\left(\sqrt{x} - 1 \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                              
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 | --------- dx = C + 2*\/ x  + 2*log\-1 + \/ x /
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 | \/ x  - 1                                     
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∫1x−1 dx=C+2x+2log⁡(x−1)\int \frac{1}{\sqrt{x} - 1}\, dx = C + 2 \sqrt{x} + 2 \log{\left(\sqrt{x} - 1 \right)}
Simplify
The graph
2.00002.01002.00102.00202.00302.00402.00502.00602.00702.00802.00900.02.5
Plot the graph f(x)
The answer [src] 
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∞\infty 
=
= 
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∞\infty 
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    Use the examples entering the upper and lower limits of integration.

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