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(x^3)/(x+3)
Solving integrals/ (x^3)/(x+3)

Integral of (x^3)/(x+3) dx

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The solution

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  1         
  /         
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 |     3    
 |    x     
 |  ----- dx
 |  x + 3   
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∫01x3x+3 dx\int\limits_{0}^{1} \frac{x^{3}}{x + 3}\, dx 
Integral(x^3/(x + 3), (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

    x3x+3=x2−3x+9−27x+3\frac{x^{3}}{x + 3} = x^{2} - 3 x + 9 - \frac{27}{x + 3}

  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:

      ∫x2 dx=x33\int x^{2}\, dx = \frac{x^{3}}{3}

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

      ∫(−3x) dx=−3∫x dx\int \left(- 3 x\right)\, dx = - 3 \int x\, dx

      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}

      So, the result is: −3x22- \frac{3 x^{2}}{2}

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

      ∫9 dx=9x\int 9\, dx = 9 x

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

      ∫(−27x+3) dx=−27∫1x+3 dx\int \left(- \frac{27}{x + 3}\right)\, dx = - 27 \int \frac{1}{x + 3}\, dx

      1. Let u=x+3u = x + 3.

        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+3)\log{\left(x + 3 \right)}

      So, the result is: −27log⁡(x+3)- 27 \log{\left(x + 3 \right)}

    The result is: x33−3x22+9x−27log⁡(x+3)\frac{x^{3}}{3} - \frac{3 x^{2}}{2} + 9 x - 27 \log{\left(x + 3 \right)}

  3. Add the constant of integration:

    x33−3x22+9x−27log⁡(x+3)+constant\frac{x^{3}}{3} - \frac{3 x^{2}}{2} + 9 x - 27 \log{\left(x + 3 \right)}+ \mathrm{constant}

The answer is:

x33−3x22+9x−27log⁡(x+3)+constant\frac{x^{3}}{3} - \frac{3 x^{2}}{2} + 9 x - 27 \log{\left(x + 3 \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                              
 |                                               
 |    3                                    2    3
 |   x                                  3*x    x 
 | ----- dx = C - 27*log(3 + x) + 9*x - ---- + --
 | x + 3                                 2     3 
 |                                               
/
2 x3−9 x2+54 x6−27 log⁡(x+3){{2\,x^3-9\,x^2+54\,x}\over{6}}-27\,\log \left(x+3\right)
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90-5050
Plot the graph f(x)
The answer [src] 
47/6 - 27*log(4) + 27*log(3)
27 log⁡3−162 log⁡4−47627\,\log 3-{{162\,\log 4-47}\over{6}} 
=
= 
47/6 - 27*log(4) + 27*log(3)
−27log⁡(4)+476+27log⁡(3)- 27 \log{\left(4 \right)} + \frac{47}{6} + 27 \log{\left(3 \right)}
Упростить
Numerical answer [src] 
0.0659173771352483
0.0659173771352483
The graph
Integral of (x^3)/(x+3) dx

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

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