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Integral of xdx/(2x+1)•(x-2) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

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  1                   
  /                   
 |                    
 |     x              
 |  -------*(x - 2) dx
 |  2*x + 1           
 |                    
/                     
0                     
$$\int\limits_{0}^{1} \frac{x}{2 x + 1} \left(x - 2\right)\, dx$$
Integral((x/(2*x + 1))*(x - 2), (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

    2. Integrate term-by-term:

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

        1. The integral of is when :

        So, the result is:

      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 a constant times a function is the constant times the integral of the function:

            1. The integral of is .

            So, the result is:

          Now substitute back in:

        So, the result is:

      The result is:

    Method #2

    1. Rewrite the integrand:

    2. Rewrite the integrand:

    3. Integrate term-by-term:

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

        1. The integral of is when :

        So, the result is:

      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 a constant times a function is the constant times the integral of the function:

            1. The integral of is .

            So, the result is:

          Now substitute back in:

        So, the result is:

      The result is:

    Method #3

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      1. Rewrite the integrand:

      2. Integrate term-by-term:

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

          1. The integral of is when :

          So, the result is:

        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 a constant times a function is the constant times the integral of the function:

              1. The integral of is .

              So, the result is:

            Now substitute back in:

          So, the result is:

        The result is:

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

        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 a constant times a function is the constant times the integral of the function:

                1. The integral of is .

                So, the result is:

              Now substitute back in:

            So, the result is:

          The result is:

        So, the result is:

      The result is:

  2. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                  
 |                                 2                 
 |    x                     5*x   x    5*log(1 + 2*x)
 | -------*(x - 2) dx = C - --- + -- + --------------
 | 2*x + 1                   4    4          8       
 |                                                   
/                                                    
$$\int \frac{x}{2 x + 1} \left(x - 2\right)\, dx = C + \frac{x^{2}}{4} - \frac{5 x}{4} + \frac{5 \log{\left(2 x + 1 \right)}}{8}$$
The graph
The answer [src]
     5*log(3)
-1 + --------
        8    
$$-1 + \frac{5 \log{\left(3 \right)}}{8}$$
=
=
     5*log(3)
-1 + --------
        8    
$$-1 + \frac{5 \log{\left(3 \right)}}{8}$$
-1 + 5*log(3)/8
Numerical answer [src]
-0.313367319582431
-0.313367319582431

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