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x^3/(x^2+1)

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

Limits of integration:

from to
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The graph:

from to

Piecewise:

The solution

You have entered [src]
  1          
  /          
 |           
 |     3     
 |    x      
 |  ------ dx
 |   2       
 |  x  + 1   
 |           
/            
0            
$$\int\limits_{0}^{1} \frac{x^{3}}{x^{2} + 1}\, dx$$
Integral(x^3/(x^2 + 1), (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Let .

      Then let and substitute :

      1. Rewrite the integrand:

      2. Integrate term-by-term:

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

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

          1. Let .

            Then let and substitute :

            1. The integral of is .

            Now substitute back in:

          So, the result is:

        The result is:

      Now substitute back in:

    Method #2

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      1. The integral of is when :

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

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

          1. Let .

            Then let and substitute :

            1. The integral of is .

            Now substitute back in:

          So, the result is:

        So, the result is:

      The result is:

  2. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                
 |                                 
 |    3             2      /     2\
 |   x             x    log\1 + x /
 | ------ dx = C + -- - -----------
 |  2              2         2     
 | x  + 1                          
 |                                 
/                                  
$$\int \frac{x^{3}}{x^{2} + 1}\, dx = C + \frac{x^{2}}{2} - \frac{\log{\left(x^{2} + 1 \right)}}{2}$$
The graph
The answer [src]
1   log(2)
- - ------
2     2   
$$\frac{1}{2} - \frac{\log{\left(2 \right)}}{2}$$
=
=
1   log(2)
- - ------
2     2   
$$\frac{1}{2} - \frac{\log{\left(2 \right)}}{2}$$
1/2 - log(2)/2
Numerical answer [src]
0.153426409720027
0.153426409720027
The graph
Integral of x^3/(x^2+1) dx

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