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x^5/(x^2+1)
Solving integrals/ x^5/(x^2+1)

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x^5/(x^2+1)

What you mean?

x^5/(x^2 + 1)
 
x^5/(x^2) + 1
 
x^5*2/x + 1
 
x^5/(x^2) + 1
 
x*5/(x^2 + 1)
 
x*5/x^2 + 1
 
x^5/(x^2) + 1

Integral of x^5/(x^2+1) dx

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The solution

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∫01x5x2+1 dx\int\limits_{0}^{1} \frac{x^{5}}{x^{2} + 1}\, dx 
Integral(x^5/(x^2 + 1), (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 dudu:

      ∫u22u+2 du\int \frac{u^{2}}{2 u + 2}\, du

      1. Rewrite the integrand:

        u22u+2=u2−12+12(u+1)\frac{u^{2}}{2 u + 2} = \frac{u}{2} - \frac{1}{2} + \frac{1}{2 \left(u + 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:

          ∫u2 du=∫u du2\int \frac{u}{2}\, du = \frac{\int u\, du}{2}

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

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

          So, the result is: u24\frac{u^{2}}{4}

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

          ∫(−12) du=−u2\int \left(- \frac{1}{2}\right)\, du = - \frac{u}{2}

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

          ∫12(u+1) du=∫1u+1 du2\int \frac{1}{2 \left(u + 1\right)}\, du = \frac{\int \frac{1}{u + 1}\, du}{2}

          1. Let u=u+1u = u + 1.

            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+1)\log{\left(u + 1 \right)}

          So, the result is: log⁡(u+1)2\frac{\log{\left(u + 1 \right)}}{2}

        The result is: u24−u2+log⁡(u+1)2\frac{u^{2}}{4} - \frac{u}{2} + \frac{\log{\left(u + 1 \right)}}{2}

      Now substitute uu back in:

      x44−x22+log⁡(x2+1)2\frac{x^{4}}{4} - \frac{x^{2}}{2} + \frac{\log{\left(x^{2} + 1 \right)}}{2}

    Method #2

    1. Rewrite the integrand:

      x5x2+1=x3−x+xx2+1\frac{x^{5}}{x^{2} + 1} = x^{3} - x + \frac{x}{x^{2} + 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:

        ∫x3 dx=x44\int x^{3}\, dx = \frac{x^{4}}{4}

      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:

        ∫xx2+1 dx=∫2xx2+1 dx2\int \frac{x}{x^{2} + 1}\, dx = \frac{\int \frac{2 x}{x^{2} + 1}\, dx}{2}

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

          Then let du=2xdxdu = 2 x dx and substitute du2\frac{du}{2}:

          ∫12u du\int \frac{1}{2 u}\, du

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

          Now substitute uu back in:

          log⁡(x2+1)\log{\left(x^{2} + 1 \right)}

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

      The result is: x44−x22+log⁡(x2+1)2\frac{x^{4}}{4} - \frac{x^{2}}{2} + \frac{\log{\left(x^{2} + 1 \right)}}{2}

  2. Add the constant of integration:

    x44−x22+log⁡(x2+1)2+constant\frac{x^{4}}{4} - \frac{x^{2}}{2} + \frac{\log{\left(x^{2} + 1 \right)}}{2}+ \mathrm{constant}

The answer is:

x44−x22+log⁡(x2+1)2+constant\frac{x^{4}}{4} - \frac{x^{2}}{2} + \frac{\log{\left(x^{2} + 1 \right)}}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                     
 |                                      
 |    5               /     2\    2    4
 |   x             log\1 + x /   x    x 
 | ------ dx = C + ----------- - -- + --
 |  2                   2        2    4 
 | x  + 1                               
 |                                      
/
log⁡(x2+1)2+x4−2 x24{{\log \left(x^2+1\right)}\over{2}}+{{x^4-2\,x^2}\over{4}}
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] 
  1   log(2)
- - + ------
  4     2
log⁡22−14{{\log 2}\over{2}}-{{1}\over{4}} 
=
= 
  1   log(2)
- - + ------
  4     2
−14+log⁡(2)2- \frac{1}{4} + \frac{\log{\left(2 \right)}}{2}
Упростить
Numerical answer [src] 
0.0965735902799727
0.0965735902799727
The graph
Integral of x^5/(x^2+1) dx

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

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