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

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

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The solution

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∫01x3x−1 dx\int\limits_{0}^{1} \frac{x^{3}}{x - 1}\, dx 
Integral(x^3/(x - 1), (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

    x3x−1=x2+x+1+1x−1\frac{x^{3}}{x - 1} = x^{2} + x + 1 + \frac{1}{x - 1}

  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:

      ∫x2 dx=x33\int x^{2}\, dx = \frac{x^{3}}{3}

    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 is the constant times the variable of integration:

      ∫1 dx=x\int 1\, dx = x

    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)}

    The result is: x33+x22+x+log⁡(x−1)\frac{x^{3}}{3} + \frac{x^{2}}{2} + x + \log{\left(x - 1 \right)}

  3. Add the constant of integration:

    x33+x22+x+log⁡(x−1)+constant\frac{x^{3}}{3} + \frac{x^{2}}{2} + x + \log{\left(x - 1 \right)}+ \mathrm{constant}

The answer is:

x33+x22+x+log⁡(x−1)+constant\frac{x^{3}}{3} + \frac{x^{2}}{2} + x + \log{\left(x - 1 \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
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 |    3                2    3              
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 | ----- dx = C + x + -- + -- + log(-1 + x)
 | x - 1              2    3               
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∫x3x−1 dx=C+x33+x22+x+log⁡(x−1)\int \frac{x^{3}}{x - 1}\, dx = C + \frac{x^{3}}{3} + \frac{x^{2}}{2} + x + \log{\left(x - 1 \right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90-1000010000
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The answer [src] 
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−∞−iπ-\infty - i \pi 
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−∞−iπ-\infty - i \pi 
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Numerical answer [src] 
-42.2576234528862
-42.2576234528862
The graph
Integral of x^3/(x-1) dx

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

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