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x^4/(x^2+1)
Solving integrals/ x^4/(x^2+1)

Integral of x^4/(x^2+1) dx

Limits of integration:

from to
v

The graph:

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Piecewise:

The solution

You have entered [src] 
  1          
  /          
 |           
 |     4     
 |    x      
 |  ------ dx
 |   2       
 |  x  + 1   
 |           
/            
0
∫01x4x2+1 dx\int\limits_{0}^{1} \frac{x^{4}}{x^{2} + 1}\, dx 
Integral(x^4/(x^2 + 1), (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

    x4x2+1=x2−1+1x2+1\frac{x^{4}}{x^{2} + 1} = x^{2} - 1 + \frac{1}{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:

      ∫x2 dx=x33\int x^{2}\, dx = \frac{x^{3}}{3}

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

      ∫(−1) dx=−x\int \left(-1\right)\, dx = - x

      PiecewiseRule(subfunctions=[(ArctanRule(a=1, b=1, c=1, context=1/(x**2 + 1), symbol=x), True), (ArccothRule(a=1, b=1, c=1, context=1/(x**2 + 1), symbol=x), False), (ArctanhRule(a=1, b=1, c=1, context=1/(x**2 + 1), symbol=x), False)], context=1/(x**2 + 1), symbol=x)

    The result is: x33−x+atan⁡(x)\frac{x^{3}}{3} - x + \operatorname{atan}{\left(x \right)}

  3. Add the constant of integration:

    x33−x+atan⁡(x)+constant\frac{x^{3}}{3} - x + \operatorname{atan}{\left(x \right)}+ \mathrm{constant}

The answer is:

x33−x+atan⁡(x)+constant\frac{x^{3}}{3} - x + \operatorname{atan}{\left(x \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                
 |                                 
 |    4                 3          
 |   x                 x           
 | ------ dx = C - x + -- + atan(x)
 |  2                  3           
 | x  + 1                          
 |                                 
/
∫x4x2+1 dx=C+x33−x+atan⁡(x)\int \frac{x^{4}}{x^{2} + 1}\, dx = C + \frac{x^{3}}{3} - x + \operatorname{atan}{\left(x \right)}
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] 
  2   pi
- - + --
  3   4
−23+π4- \frac{2}{3} + \frac{\pi}{4} 
=
= 
  2   pi
- - + --
  3   4
−23+π4- \frac{2}{3} + \frac{\pi}{4} 
-2/3 + pi/4
Numerical answer [src] 
0.118731496730782
0.118731496730782
The graph
Integral of x^4/(x^2+1) dx

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

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