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

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0111x+13dx\int\limits_{0}^{1} \frac{1}{1 - \sqrt[3]{x + 1}}\, dx
Integral(1/(1 - (x + 1)^(1/3)), (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=x+13u = \sqrt[3]{x + 1}.

      Then let du=dx3(x+1)23du = \frac{dx}{3 \left(x + 1\right)^{\frac{2}{3}}} and substitute 3du- 3 du:

      (3u2u1)du\int \left(- \frac{3 u^{2}}{u - 1}\right)\, du

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

        u2u1du=3u2u1du\int \frac{u^{2}}{u - 1}\, du = - 3 \int \frac{u^{2}}{u - 1}\, du

        1. Rewrite the integrand:

          u2u1=u+1+1u1\frac{u^{2}}{u - 1} = u + 1 + \frac{1}{u - 1}

        2. Integrate term-by-term:

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n1n \neq -1:

            udu=u22\int u\, du = \frac{u^{2}}{2}

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

            1du=u\int 1\, du = u

          1. Let u=u1u = u - 1.

            Then let du=dudu = du and substitute dudu:

            1udu\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(u1)\log{\left(u - 1 \right)}

          The result is: u22+u+log(u1)\frac{u^{2}}{2} + u + \log{\left(u - 1 \right)}

        So, the result is: 3u223u3log(u1)- \frac{3 u^{2}}{2} - 3 u - 3 \log{\left(u - 1 \right)}

      Now substitute uu back in:

      3(x+1)2323x+133log(x+131)- \frac{3 \left(x + 1\right)^{\frac{2}{3}}}{2} - 3 \sqrt[3]{x + 1} - 3 \log{\left(\sqrt[3]{x + 1} - 1 \right)}

    Method #2

    1. Rewrite the integrand:

      11x+13=1x+131\frac{1}{1 - \sqrt[3]{x + 1}} = - \frac{1}{\sqrt[3]{x + 1} - 1}

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

      (1x+131)dx=1x+131dx\int \left(- \frac{1}{\sqrt[3]{x + 1} - 1}\right)\, dx = - \int \frac{1}{\sqrt[3]{x + 1} - 1}\, dx

      1. Let u=x+13u = \sqrt[3]{x + 1}.

        Then let du=dx3(x+1)23du = \frac{dx}{3 \left(x + 1\right)^{\frac{2}{3}}} and substitute 3du3 du:

        3u2u1du\int \frac{3 u^{2}}{u - 1}\, du

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

          u2u1du=3u2u1du\int \frac{u^{2}}{u - 1}\, du = 3 \int \frac{u^{2}}{u - 1}\, du

          1. Rewrite the integrand:

            u2u1=u+1+1u1\frac{u^{2}}{u - 1} = u + 1 + \frac{1}{u - 1}

          2. Integrate term-by-term:

            1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n1n \neq -1:

              udu=u22\int u\, du = \frac{u^{2}}{2}

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

              1du=u\int 1\, du = u

            1. Let u=u1u = u - 1.

              Then let du=dudu = du and substitute dudu:

              1udu\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(u1)\log{\left(u - 1 \right)}

            The result is: u22+u+log(u1)\frac{u^{2}}{2} + u + \log{\left(u - 1 \right)}

          So, the result is: 3u22+3u+3log(u1)\frac{3 u^{2}}{2} + 3 u + 3 \log{\left(u - 1 \right)}

        Now substitute uu back in:

        3(x+1)232+3x+13+3log(x+131)\frac{3 \left(x + 1\right)^{\frac{2}{3}}}{2} + 3 \sqrt[3]{x + 1} + 3 \log{\left(\sqrt[3]{x + 1} - 1 \right)}

      So, the result is: 3(x+1)2323x+133log(x+131)- \frac{3 \left(x + 1\right)^{\frac{2}{3}}}{2} - 3 \sqrt[3]{x + 1} - 3 \log{\left(\sqrt[3]{x + 1} - 1 \right)}

  2. Now simplify:

    3(x+1)2323x+133log(x+131)- \frac{3 \left(x + 1\right)^{\frac{2}{3}}}{2} - 3 \sqrt[3]{x + 1} - 3 \log{\left(\sqrt[3]{x + 1} - 1 \right)}

  3. Add the constant of integration:

    3(x+1)2323x+133log(x+131)+constant- \frac{3 \left(x + 1\right)^{\frac{2}{3}}}{2} - 3 \sqrt[3]{x + 1} - 3 \log{\left(\sqrt[3]{x + 1} - 1 \right)}+ \mathrm{constant}


The answer is:

3(x+1)2323x+133log(x+131)+constant- \frac{3 \left(x + 1\right)^{\frac{2}{3}}}{2} - 3 \sqrt[3]{x + 1} - 3 \log{\left(\sqrt[3]{x + 1} - 1 \right)}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                                                                         
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 |       1                  3 _______        /     3 _______\   3*(x + 1)   
 | ------------- dx = C - 3*\/ x + 1  - 3*log\-1 + \/ x + 1 / - ------------
 |     3 _______                                                     2      
 | 1 - \/ x + 1                                                             
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11x+13dx=C3(x+1)2323x+133log(x+131)\int \frac{1}{1 - \sqrt[3]{x + 1}}\, dx = C - \frac{3 \left(x + 1\right)^{\frac{2}{3}}}{2} - 3 \sqrt[3]{x + 1} - 3 \log{\left(\sqrt[3]{x + 1} - 1 \right)}
The graph
0.001.000.100.200.300.400.500.600.700.800.90-5000050000
The answer [src]
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-\infty
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-\infty
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Numerical answer [src]
-133.185682576228
-133.185682576228

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