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

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

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The solution

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∫01x2x3+1 dx\int\limits_{0}^{1} \frac{x^{2}}{x^{3} + 1}\, dx 
Integral(x^2/(1 + x^3), (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=x3+1u = x^{3} + 1.

      Then let du=3x2dxdu = 3 x^{2} dx and substitute du3\frac{du}{3}:

      ∫13u du\int \frac{1}{3 u}\, du

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

        ∫1u du=∫1u du3\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{3}

        1. The integral of 1u\frac{1}{u} is log⁡(u)\log{\left(u \right)}.

        So, the result is: log⁡(u)3\frac{\log{\left(u \right)}}{3}

      Now substitute uu back in:

      log⁡(x3+1)3\frac{\log{\left(x^{3} + 1 \right)}}{3}

    Method #2

    1. Rewrite the integrand:

      x2x3+1=2x−13(x2−x+1)+13(x+1)\frac{x^{2}}{x^{3} + 1} = \frac{2 x - 1}{3 \left(x^{2} - x + 1\right)} + \frac{1}{3 \left(x + 1\right)}

    2. Integrate term-by-term:

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

        ∫2x−13(x2−x+1) dx=∫2x−1x2−x+1 dx3\int \frac{2 x - 1}{3 \left(x^{2} - x + 1\right)}\, dx = \frac{\int \frac{2 x - 1}{x^{2} - x + 1}\, dx}{3}

        1. Let u=x2−x+1u = x^{2} - x + 1.

          Then let du=(2x−1)dxdu = \left(2 x - 1\right) 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⁡(x2−x+1)\log{\left(x^{2} - x + 1 \right)}

        So, the result is: log⁡(x2−x+1)3\frac{\log{\left(x^{2} - x + 1 \right)}}{3}

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

        ∫13(x+1) dx=∫1x+1 dx3\int \frac{1}{3 \left(x + 1\right)}\, dx = \frac{\int \frac{1}{x + 1}\, dx}{3}

        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)3\frac{\log{\left(x + 1 \right)}}{3}

      The result is: log⁡(x+1)3+log⁡(x2−x+1)3\frac{\log{\left(x + 1 \right)}}{3} + \frac{\log{\left(x^{2} - x + 1 \right)}}{3}

  2. Add the constant of integration:

    log⁡(x3+1)3+constant\frac{\log{\left(x^{3} + 1 \right)}}{3}+ \mathrm{constant}

The answer is:

log⁡(x3+1)3+constant\frac{\log{\left(x^{3} + 1 \right)}}{3}+ \mathrm{constant}

The answer (Indefinite) [src] 
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∫x2x3+1 dx=C+log⁡(x3+1)3\int \frac{x^{2}}{x^{3} + 1}\, dx = C + \frac{\log{\left(x^{3} + 1 \right)}}{3}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.01.0
Plot the graph f(x)
The answer [src] 
log(2)
------
  3
log⁡(2)3\frac{\log{\left(2 \right)}}{3} 
=
= 
log(2)
------
  3
log⁡(2)3\frac{\log{\left(2 \right)}}{3} 
log(2)/3
Numerical answer [src] 
0.231049060186648
0.231049060186648
The graph
Integral of x^2/(1+x^3) dx

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

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