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

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

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The solution

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∫01x2−1x+1 dx\int\limits_{0}^{1} \frac{x^{2} - 1}{x + 1}\, dx 
Integral((x^2 - 1)/(x + 1), (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

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

        ∫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 \left(-1\right)\, dx = - x

      The result is: x22−x\frac{x^{2}}{2} - x

    Method #2

    1. Rewrite the integrand:

      x2−1x+1=x2x+1−1x+1\frac{x^{2} - 1}{x + 1} = \frac{x^{2}}{x + 1} - \frac{1}{x + 1}

    2. Integrate term-by-term:

      1. Rewrite the integrand:

        x2x+1=x−1+1x+1\frac{x^{2}}{x + 1} = 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:

          ∫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 \left(-1\right)\, 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: x22−x+log⁡(x+1)\frac{x^{2}}{2} - x + \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. 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)- \log{\left(x + 1 \right)}

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

  2. Now simplify:

    x(x−2)2\frac{x \left(x - 2\right)}{2}

  3. Add the constant of integration:

    x(x−2)2+constant\frac{x \left(x - 2\right)}{2}+ \mathrm{constant}

The answer is:

x(x−2)2+constant\frac{x \left(x - 2\right)}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
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∫x2−1x+1 dx=C+x22−x\int \frac{x^{2} - 1}{x + 1}\, dx = C + \frac{x^{2}}{2} - x
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.901-2
Plot the graph f(x)
The answer [src] 
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−12- \frac{1}{2} 
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−12- \frac{1}{2} 
-1/2
Numerical answer [src] 
-0.5
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The graph
Integral of (x^2-1)/(x+1) dx

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

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