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(x^3)/(4-x^2)
Solving integrals/ (x^3)/(4-x^2)

Integral of (x^3)/(4-x^2) dx

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∫01x34−x2 dx\int\limits_{0}^{1} \frac{x^{3}}{4 - x^{2}}\, dx 
Integral(x^3/(4 - x^2), (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 −du- du:

      ∫(−u2u−8) du\int \left(- \frac{u}{2 u - 8}\right)\, du

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

        ∫u2u−8 du=−∫u2u−8 du\int \frac{u}{2 u - 8}\, du = - \int \frac{u}{2 u - 8}\, du

        1. Rewrite the integrand:

          u2u−8=12+2u−4\frac{u}{2 u - 8} = \frac{1}{2} + \frac{2}{u - 4}

        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:

            ∫2u−4 du=2∫1u−4 du\int \frac{2}{u - 4}\, du = 2 \int \frac{1}{u - 4}\, du

            1. Let u=u−4u = u - 4.

              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−4)\log{\left(u - 4 \right)}

            So, the result is: 2log⁡(u−4)2 \log{\left(u - 4 \right)}

          The result is: u2+2log⁡(u−4)\frac{u}{2} + 2 \log{\left(u - 4 \right)}

        So, the result is: −u2−2log⁡(u−4)- \frac{u}{2} - 2 \log{\left(u - 4 \right)}

      Now substitute uu back in:

      −x22−2log⁡(x2−4)- \frac{x^{2}}{2} - 2 \log{\left(x^{2} - 4 \right)}

    Method #2

    1. Rewrite the integrand:

      x34−x2=−x−2x+2−2x−2\frac{x^{3}}{4 - x^{2}} = - x - \frac{2}{x + 2} - \frac{2}{x - 2}

    2. Integrate term-by-term:

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

        ∫(−x) dx=−∫x dx\int \left(- x\right)\, dx = - \int x\, dx

        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}

        So, the result is: −x22- \frac{x^{2}}{2}

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

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

        1. Let u=x+2u = x + 2.

          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+2)\log{\left(x + 2 \right)}

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

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

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

        1. Let u=x−2u = x - 2.

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

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

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

    Method #3

    1. Rewrite the integrand:

      x34−x2=−x3x2−4\frac{x^{3}}{4 - x^{2}} = - \frac{x^{3}}{x^{2} - 4}

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

      ∫(−x3x2−4) dx=−∫x3x2−4 dx\int \left(- \frac{x^{3}}{x^{2} - 4}\right)\, dx = - \int \frac{x^{3}}{x^{2} - 4}\, dx

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

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

        ∫u2u−8 du\int \frac{u}{2 u - 8}\, du

        1. Rewrite the integrand:

          u2u−8=12+2u−4\frac{u}{2 u - 8} = \frac{1}{2} + \frac{2}{u - 4}

        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:

            ∫2u−4 du=2∫1u−4 du\int \frac{2}{u - 4}\, du = 2 \int \frac{1}{u - 4}\, du

            1. Let u=u−4u = u - 4.

              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−4)\log{\left(u - 4 \right)}

            So, the result is: 2log⁡(u−4)2 \log{\left(u - 4 \right)}

          The result is: u2+2log⁡(u−4)\frac{u}{2} + 2 \log{\left(u - 4 \right)}

        Now substitute uu back in:

        x22+2log⁡(x2−4)\frac{x^{2}}{2} + 2 \log{\left(x^{2} - 4 \right)}

      So, the result is: −x22−2log⁡(x2−4)- \frac{x^{2}}{2} - 2 \log{\left(x^{2} - 4 \right)}

  2. Add the constant of integration:

    −x22−2log⁡(x2−4)+constant- \frac{x^{2}}{2} - 2 \log{\left(x^{2} - 4 \right)}+ \mathrm{constant}

The answer is:

−x22−2log⁡(x2−4)+constant- \frac{x^{2}}{2} - 2 \log{\left(x^{2} - 4 \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                   
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 | ------ dx = C - 2*log\-4 + x / - --
 |      2                           2 
 | 4 - x                              
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∫x34−x2 dx=C−x22−2log⁡(x2−4)\int \frac{x^{3}}{4 - x^{2}}\, dx = C - \frac{x^{2}}{2} - 2 \log{\left(x^{2} - 4 \right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.00.5
Plot the graph f(x)
The answer [src] 
-1/2 - 2*log(3) + 2*log(4)
−2log⁡(3)−12+2log⁡(4)- 2 \log{\left(3 \right)} - \frac{1}{2} + 2 \log{\left(4 \right)} 
=
= 
-1/2 - 2*log(3) + 2*log(4)
−2log⁡(3)−12+2log⁡(4)- 2 \log{\left(3 \right)} - \frac{1}{2} + 2 \log{\left(4 \right)} 
-1/2 - 2*log(3) + 2*log(4)
Numerical answer [src] 
0.0753641449035619
0.0753641449035619
The graph
Integral of (x^3)/(4-x^2) dx

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

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