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

Integral of x^2/(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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∫01x2x+2 dx\int\limits_{0}^{1} \frac{x^{2}}{x + 2}\, dx 
Integral(x^2/(x + 2), (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

    x2x+2=x−2+4x+2\frac{x^{2}}{x + 2} = x - 2 + \frac{4}{x + 2}

  2. Integrate term-by-term:

    1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

      ∫x dx=x22\int x\, dx = \frac{x^{2}}{2}

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

      ∫(−2) dx=−2x\int \left(-2\right)\, 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 \frac{4}{x + 2}\, 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: x22−2x+4log⁡(x+2)\frac{x^{2}}{2} - 2 x + 4 \log{\left(x + 2 \right)}

  3. Add the constant of integration:

    x22−2x+4log⁡(x+2)+constant\frac{x^{2}}{2} - 2 x + 4 \log{\left(x + 2 \right)}+ \mathrm{constant}

The answer is:

x22−2x+4log⁡(x+2)+constant\frac{x^{2}}{2} - 2 x + 4 \log{\left(x + 2 \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
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 | ----- dx = C + -- - 2*x + 4*log(2 + x)
 | x + 2          2                      
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∫x2x+2 dx=C+x22−2x+4log⁡(x+2)\int \frac{x^{2}}{x + 2}\, dx = C + \frac{x^{2}}{2} - 2 x + 4 \log{\left(x + 2 \right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.9005
Plot the graph f(x)
The answer [src] 
-3/2 - 4*log(2) + 4*log(3)
−4log⁡(2)−32+4log⁡(3)- 4 \log{\left(2 \right)} - \frac{3}{2} + 4 \log{\left(3 \right)} 
=
= 
-3/2 - 4*log(2) + 4*log(3)
−4log⁡(2)−32+4log⁡(3)- 4 \log{\left(2 \right)} - \frac{3}{2} + 4 \log{\left(3 \right)} 
-3/2 - 4*log(2) + 4*log(3)
Numerical answer [src] 
0.121860432432658
0.121860432432658
The graph
Integral of x^2/(x+2) dx

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

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