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Solving integrals/ 2*x/(x+2)

Integral of 2*x/(x+2) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
  2         
  /         
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 |   2*x    
 |  ----- dx
 |  x + 2   
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0
∫022xx+2 dx\int\limits_{0}^{2} \frac{2 x}{x + 2}\, dx 
Integral((2*x)/(x + 2), (x, 0, 2))
Detail solution

  1. Rewrite the integrand:

    2xx+2=2−4x+2\frac{2 x}{x + 2} = 2 - \frac{4}{x + 2}

  2. Integrate term-by-term:

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

      ∫2 dx=2x\int 2\, dx = 2 x

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

      ∫(−4x+2) dx=−4∫1x+2 dx\int \left(- \frac{4}{x + 2}\right)\, dx = - 4 \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: −4log⁡(x+2)- 4 \log{\left(x + 2 \right)}

    The result is: 2x−4log⁡(x+2)2 x - 4 \log{\left(x + 2 \right)}

  3. Add the constant of integration:

    2x−4log⁡(x+2)+constant2 x - 4 \log{\left(x + 2 \right)}+ \mathrm{constant}

The answer is:

2x−4log⁡(x+2)+constant2 x - 4 \log{\left(x + 2 \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                 
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 |  2*x                             
 | ----- dx = C - 4*log(2 + x) + 2*x
 | x + 2                            
 |                                  
/
∫2xx+2 dx=C+2x−4log⁡(x+2)\int \frac{2 x}{x + 2}\, dx = C + 2 x - 4 \log{\left(x + 2 \right)}
Simplify
The graph
0.02.00.20.40.60.81.01.21.41.61.85-5
Plot the graph f(x)
The answer [src] 
4 - 4*log(4) + 4*log(2)
−4log⁡(4)+4log⁡(2)+4- 4 \log{\left(4 \right)} + 4 \log{\left(2 \right)} + 4 
=
= 
4 - 4*log(4) + 4*log(2)
−4log⁡(4)+4log⁡(2)+4- 4 \log{\left(4 \right)} + 4 \log{\left(2 \right)} + 4 
4 - 4*log(4) + 4*log(2)
Numerical answer [src] 
1.22741127776022
1.22741127776022

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

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