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

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

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The solution

You have entered [src] 
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 |  /3 ___       1   \   
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∫01(x3−1⋅1x2−1) dx\int\limits_{0}^{1} \left(\sqrt[3]{x} - 1 \cdot \frac{1}{x^{2} - 1}\right)\, dx 
Integral(x^(1/3) - 1/(x^2 - 1*1), (x, 0, 1))
Detail solution

  1. Integrate term-by-term:

    1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

      ∫x3 dx=3x434\int \sqrt[3]{x}\, dx = \frac{3 x^{\frac{4}{3}}}{4}

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

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

      1. Rewrite the integrand:

        1x2−1=−1x+1+1x−12\frac{1}{x^{2} - 1} = \frac{- \frac{1}{x + 1} + \frac{1}{x - 1}}{2}

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

        ∫−1x+1+1x−12 dx=∫(−1x+1+1x−1) dx2\int \frac{- \frac{1}{x + 1} + \frac{1}{x - 1}}{2}\, dx = \frac{\int \left(- \frac{1}{x + 1} + \frac{1}{x - 1}\right)\, dx}{2}

        1. Integrate term-by-term:

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

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

            ∫(−1x+1) dx=−∫1x+1 dx\int \left(- \frac{1}{x + 1}\right)\, dx = - \int \frac{1}{x + 1}\, dx

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

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

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

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

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

    The result is: 3x434−log⁡(x−1)2+log⁡(x+1)2\frac{3 x^{\frac{4}{3}}}{4} - \frac{\log{\left(x - 1 \right)}}{2} + \frac{\log{\left(x + 1 \right)}}{2}

  2. Add the constant of integration:

    3x434−log⁡(x−1)2+log⁡(x+1)2+constant\frac{3 x^{\frac{4}{3}}}{4} - \frac{\log{\left(x - 1 \right)}}{2} + \frac{\log{\left(x + 1 \right)}}{2}+ \mathrm{constant}

The answer is:

3x434−log⁡(x−1)2+log⁡(x+1)2+constant\frac{3 x^{\frac{4}{3}}}{4} - \frac{\log{\left(x - 1 \right)}}{2} + \frac{\log{\left(x + 1 \right)}}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                             
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 | /3 ___       1   \          log(1 + x)   log(-1 + x)   3*x   
 | |\/ x  - 1*------| dx = C + ---------- - ----------- + ------
 | |           2    |              2             2          4   
 | \          x  - 1/                                           
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/
log⁡(x+1)2+3 x434−log⁡(x−1)2{{\log \left(x+1\right)}\over{2}}+{{3\,x^{{{4}\over{3}}}}\over{4}}- {{\log \left(x-1\right)}\over{2}}
Simplify
The answer [src] 
     pi*I
oo + ----
      2
%a{\it \%a} 
=
= 
     pi*I
oo + ----
      2
∞+iπ2\infty + \frac{i \pi}{2}
Simplify
Numerical answer [src] 
23.1420519833869
23.1420519833869

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

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