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

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

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The solution

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∫1∞1x2(x−1) dx\int\limits_{1}^{\infty} \frac{1}{x^{2} \left(\sqrt{x} - 1\right)}\, dx 
Integral(1/((sqrt(x) - 1)*x^2), (x, 1, oo))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

      1x2(x−1)=−1−x52+x2\frac{1}{x^{2} \left(\sqrt{x} - 1\right)} = - \frac{1}{- x^{\frac{5}{2}} + x^{2}}

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

      ∫(−1−x52+x2) dx=−∫1−x52+x2 dx\int \left(- \frac{1}{- x^{\frac{5}{2}} + x^{2}}\right)\, dx = - \int \frac{1}{- x^{\frac{5}{2}} + x^{2}}\, dx

      1. Don't know the steps in finding this integral.

        But the integral is

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

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

    Method #2

    1. Rewrite the integrand:

      1x2(x−1)=1x52−x2\frac{1}{x^{2} \left(\sqrt{x} - 1\right)} = \frac{1}{x^{\frac{5}{2}} - x^{2}}

    2. Rewrite the integrand:

      1x52−x2=−1−x52+x2\frac{1}{x^{\frac{5}{2}} - x^{2}} = - \frac{1}{- x^{\frac{5}{2}} + x^{2}}

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

      ∫(−1−x52+x2) dx=−∫1−x52+x2 dx\int \left(- \frac{1}{- x^{\frac{5}{2}} + x^{2}}\right)\, dx = - \int \frac{1}{- x^{\frac{5}{2}} + x^{2}}\, dx

      1. Don't know the steps in finding this integral.

        But the integral is

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

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

  2. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
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 | \\/ x  - 1/*x                        \/ x                     
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∫1x2(x−1) dx=C−log⁡(x)+2log⁡(x−1)+1x+2x\int \frac{1}{x^{2} \left(\sqrt{x} - 1\right)}\, dx = C - \log{\left(x \right)} + 2 \log{\left(\sqrt{x} - 1 \right)} + \frac{1}{x} + \frac{2}{\sqrt{x}}
Simplify
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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