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Integral of x/(x-1)(x-2)^2 dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

You have entered [src]
  1                  
  /                  
 |                   
 |    x          2   
 |  -----*(x - 2)  dx
 |  x - 1            
 |                   
/                    
0                    
$$\int\limits_{0}^{1} \frac{x}{x - 1} \left(x - 2\right)^{2}\, dx$$
Integral((x/(x - 1))*(x - 2)^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 is when :

      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. Let .

        Then let and substitute :

        1. The integral of is .

        Now substitute back in:

      The result is:

    Method #2

    1. Rewrite the integrand:

    2. Rewrite the integrand:

    3. 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 is when :

        So, the result is:

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

      1. Let .

        Then let and substitute :

        1. The integral of is .

        Now substitute back in:

      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 is when :

        1. The integral of is when :

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

        1. Let .

          Then let and substitute :

          1. The integral of is .

          Now substitute back in:

        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 is when :

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

          1. Let .

            Then let and substitute :

            1. The integral of is .

            Now substitute back in:

          The result is:

        So, 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. Let .

            Then let and substitute :

            1. The integral of is .

            Now substitute back in:

          The result is:

        So, the result is:

      The result is:

  2. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                   
 |                                2    3              
 |   x          2              3*x    x               
 | -----*(x - 2)  dx = C + x - ---- + -- + log(-1 + x)
 | x - 1                        2     3               
 |                                                    
/                                                     
$$\int \frac{x}{x - 1} \left(x - 2\right)^{2}\, dx = C + \frac{x^{3}}{3} - \frac{3 x^{2}}{2} + x + \log{\left(x - 1 \right)}$$
The graph
The answer [src]
-oo - pi*I
$$-\infty - i \pi$$
=
=
-oo - pi*I
$$-\infty - i \pi$$
-oo - pi*i
Numerical answer [src]
-44.2576234528862
-44.2576234528862

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