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x/(x-2)
Solving integrals/ x/(x-2)

Integral of x/(x-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         
  /         
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 |    x     
 |  ----- dx
 |  x - 2   
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0
∫01xx−2 dx\int\limits_{0}^{1} \frac{x}{x - 2}\, dx 
Integral(x/(x - 1*2), (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

    xx−2=1+2x−2\frac{x}{x - 2} = 1 + \frac{2}{x - 2}

  2. Integrate term-by-term:

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

      ∫1 dx=x\int 1\, dx = x

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

      ∫2x−2 dx=2∫1x−2 dx\int \frac{2}{x - 2}\, dx = 2 \int \frac{1}{x - 2}\, dx

      1. Let u=x−2u = x - 2.

        Then let du=dxdu = dx 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⁡(x−2)\log{\left(x - 2 \right)}

      So, the result is: 2log⁡(x−2)2 \log{\left(x - 2 \right)}

    The result is: x+2log⁡(x−2)x + 2 \log{\left(x - 2 \right)}

  3. Add the constant of integration:

    x+2log⁡(x−2)+constantx + 2 \log{\left(x - 2 \right)}+ \mathrm{constant}

The answer is:

x+2log⁡(x−2)+constantx + 2 \log{\left(x - 2 \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                
 |                                 
 |   x                             
 | ----- dx = C + x + 2*log(-2 + x)
 | x - 2                           
 |                                 
/
x+2 log⁡(x−2)x+2\,\log \left(x-2\right)
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.901-2
Plot the graph f(x)
The answer [src] 
1 - 2*log(2)
1−2 log⁡21-2\,\log 2 
=
= 
1 - 2*log(2)
−2log⁡(2)+1- 2 \log{\left(2 \right)} + 1
Simplify
Numerical answer [src] 
-0.386294361119891
-0.386294361119891
The graph
Integral of x/(x-2) dx

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

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