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x^3/(x^2-1)
Solving integrals/ x^3/(x^2-1)

Integral of x^3/(x^2-1) dx

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The solution

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∫01x3x2−1 dx\int\limits_{0}^{1} \frac{x^{3}}{x^{2} - 1}\, dx 
Integral(x^3/(x^2 - 1), (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=x2u = x^{2}.

      Then let du=2xdxdu = 2 x dx and substitute dudu:

      ∫u2u−2 du\int \frac{u}{2 u - 2}\, du

      1. Rewrite the integrand:

        u2u−2=12+12(u−1)\frac{u}{2 u - 2} = \frac{1}{2} + \frac{1}{2 \left(u - 1\right)}

      2. Integrate term-by-term:

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

          ∫12 du=u2\int \frac{1}{2}\, du = \frac{u}{2}

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

          ∫12(u−1) du=∫1u−1 du2\int \frac{1}{2 \left(u - 1\right)}\, du = \frac{\int \frac{1}{u - 1}\, du}{2}

          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)}

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

        The result is: u2+log⁡(u−1)2\frac{u}{2} + \frac{\log{\left(u - 1 \right)}}{2}

      Now substitute uu back in:

      x22+log⁡(x2−1)2\frac{x^{2}}{2} + \frac{\log{\left(x^{2} - 1 \right)}}{2}

    Method #2

    1. Rewrite the integrand:

      x3x2−1=x+12(x+1)+12(x−1)\frac{x^{3}}{x^{2} - 1} = x + \frac{1}{2 \left(x + 1\right)} + \frac{1}{2 \left(x - 1\right)}

    2. 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:

        ∫x dx=x22\int x\, dx = \frac{x^{2}}{2}

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

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

        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: log⁡(x+1)2\frac{\log{\left(x + 1 \right)}}{2}

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

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

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

      The result is: x22+log⁡(x−1)2+log⁡(x+1)2\frac{x^{2}}{2} + \frac{\log{\left(x - 1 \right)}}{2} + \frac{\log{\left(x + 1 \right)}}{2}

  2. Add the constant of integration:

    x22+log⁡(x2−1)2+constant\frac{x^{2}}{2} + \frac{\log{\left(x^{2} - 1 \right)}}{2}+ \mathrm{constant}

The answer is:

x22+log⁡(x2−1)2+constant\frac{x^{2}}{2} + \frac{\log{\left(x^{2} - 1 \right)}}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                 
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 |    3             2      /      2\
 |   x             x    log\-1 + x /
 | ------ dx = C + -- + ------------
 |  2              2         2      
 | x  - 1                           
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/
∫x3x2−1 dx=C+x22+log⁡(x2−1)2\int \frac{x^{3}}{x^{2} - 1}\, dx = C + \frac{x^{2}}{2} + \frac{\log{\left(x^{2} - 1 \right)}}{2}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90-50005000
Plot the graph f(x)
The answer [src] 
      pi*I
-oo - ----
       2
−∞−iπ2-\infty - \frac{i \pi}{2} 
=
= 
      pi*I
-oo - ----
       2
−∞−iπ2-\infty - \frac{i \pi}{2} 
-oo - pi*i/2
Numerical answer [src] 
-21.1989048028269
-21.1989048028269
The graph
Integral of x^3/(x^2-1) dx

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

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