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Solving integrals/ (x(6-x))/2

Integral of (x(6-x))/2 dx

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The solution

You have entered [src] 
  1             
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 |  x*(6 - x)   
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∫01x(6−x)2 dx\int\limits_{0}^{1} \frac{x \left(6 - x\right)}{2}\, dx 
Integral((x*(6 - x))/2, (x, 0, 1))
Detail solution

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

    ∫x(6−x)2 dx=∫x(6−x) dx2\int \frac{x \left(6 - x\right)}{2}\, dx = \frac{\int x \left(6 - x\right)\, dx}{2}

    1. There are multiple ways to do this integral.

      Method #1

      1. Let u=−xu = - x.

        Then let du=−dxdu = - dx and substitute dudu:

        ∫(u2+6u) du\int \left(u^{2} + 6 u\right)\, du

        1. Integrate term-by-term:

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u2 du=u33\int u^{2}\, du = \frac{u^{3}}{3}

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

            ∫6u du=6∫u du\int 6 u\, du = 6 \int u\, du

            1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

              ∫u du=u22\int u\, du = \frac{u^{2}}{2}

            So, the result is: 3u23 u^{2}

          The result is: u33+3u2\frac{u^{3}}{3} + 3 u^{2}

        Now substitute uu back in:

        −x33+3x2- \frac{x^{3}}{3} + 3 x^{2}

      Method #2

      1. Rewrite the integrand:

        x(6−x)=−x2+6xx \left(6 - x\right) = - x^{2} + 6 x

      2. Integrate term-by-term:

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

          ∫(−x2) dx=−∫x2 dx\int \left(- x^{2}\right)\, dx = - \int x^{2}\, dx

          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}

          So, the result is: −x33- \frac{x^{3}}{3}

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

          ∫6x dx=6∫x dx\int 6 x\, dx = 6 \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: 3x23 x^{2}

        The result is: −x33+3x2- \frac{x^{3}}{3} + 3 x^{2}

    So, the result is: −x36+3x22- \frac{x^{3}}{6} + \frac{3 x^{2}}{2}

  2. Now simplify:

    x2(9−x)6\frac{x^{2} \left(9 - x\right)}{6}

  3. Add the constant of integration:

    x2(9−x)6+constant\frac{x^{2} \left(9 - x\right)}{6}+ \mathrm{constant}

The answer is:

x2(9−x)6+constant\frac{x^{2} \left(9 - x\right)}{6}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                            
 |                     3      2
 | x*(6 - x)          x    3*x 
 | --------- dx = C - -- + ----
 |     2              6     2  
 |                             
/
∫x(6−x)2 dx=C−x36+3x22\int \frac{x \left(6 - x\right)}{2}\, dx = C - \frac{x^{3}}{6} + \frac{3 x^{2}}{2}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.05.0
Plot the graph f(x)
The answer [src] 
4/3
43\frac{4}{3} 
=
= 
4/3
43\frac{4}{3} 
4/3
Numerical answer [src] 
1.33333333333333
1.33333333333333

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

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