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Solving integrals/ (1/x+2/(x-1))dx

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

Limits of integration:

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The solution

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∫01(2x−1+1x) dx\int\limits_{0}^{1} \left(\frac{2}{x - 1} + \frac{1}{x}\right)\, dx 
Integral(1/x + 2/(x - 1), (x, 0, 1))
Detail solution

  1. Integrate term-by-term:

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

      ∫2x−1 dx=2∫1x−1 dx\int \frac{2}{x - 1}\, dx = 2 \int \frac{1}{x - 1}\, dx

      1. Let u=x−1u = x - 1.

        Then let du=dxdu = dx 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⁡(x−1)\log{\left(x - 1 \right)}

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

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

    The result is: log⁡(x)+2log⁡(x−1)\log{\left(x \right)} + 2 \log{\left(x - 1 \right)}

  2. Now simplify:

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

  3. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
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∫(2x−1+1x) dx=C+log⁡(x)+2log⁡(x−1)\int \left(\frac{2}{x - 1} + \frac{1}{x}\right)\, dx = C + \log{\left(x \right)} + 2 \log{\left(x - 1 \right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90-2000020000
Plot the graph f(x)
The answer [src] 
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Numerical answer [src] 
-44.0914674384461
-44.0914674384461

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

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