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

Integral of x/(x+2) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
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∫01xx+2 dx\int\limits_{0}^{1} \frac{x}{x + 2}\, dx 
Integral(x/(x + 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 \left(- \frac{2}{x + 2}\right)\, 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] 
  /                               
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 | ----- dx = C + x - 2*log(2 + x)
 | x + 2                          
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∫xx+2 dx=C+x−2log⁡(x+2)\int \frac{x}{x + 2}\, dx = C + x - 2 \log{\left(x + 2 \right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.902-2
Plot the graph f(x)
The answer [src] 
1 - 2*log(3) + 2*log(2)
−2log⁡(3)+1+2log⁡(2)- 2 \log{\left(3 \right)} + 1 + 2 \log{\left(2 \right)} 
=
= 
1 - 2*log(3) + 2*log(2)
−2log⁡(3)+1+2log⁡(2)- 2 \log{\left(3 \right)} + 1 + 2 \log{\left(2 \right)} 
1 - 2*log(3) + 2*log(2)
Numerical answer [src] 
0.189069783783671
0.189069783783671
The graph
Integral of x/(x+2) dx

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

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