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x(x-1)(x-2)

Integral of x(x-1)(x-2) dx

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The solution

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01x(x1)(x2)dx\int\limits_{0}^{1} x \left(x - 1\right) \left(x - 2\right)\, dx
Integral((x*(x - 1))*(x - 2), (x, 0, 1))
Detail solution
  1. Rewrite the integrand:

    x(x1)(x2)=x33x2+2xx \left(x - 1\right) \left(x - 2\right) = x^{3} - 3 x^{2} + 2 x

  2. Integrate term-by-term:

    1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n1n \neq -1:

      x3dx=x44\int x^{3}\, dx = \frac{x^{4}}{4}

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

      (3x2)dx=3x2dx\int \left(- 3 x^{2}\right)\, dx = - 3 \int x^{2}\, dx

      1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n1n \neq -1:

        x2dx=x33\int x^{2}\, dx = \frac{x^{3}}{3}

      So, the result is: x3- x^{3}

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

      2xdx=2xdx\int 2 x\, dx = 2 \int x\, dx

      1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n1n \neq -1:

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

      So, the result is: x2x^{2}

    The result is: x44x3+x2\frac{x^{4}}{4} - x^{3} + x^{2}

  3. Now simplify:

    x2(x24x+1)x^{2} \left(\frac{x^{2}}{4} - x + 1\right)

  4. Add the constant of integration:

    x2(x24x+1)+constantx^{2} \left(\frac{x^{2}}{4} - x + 1\right)+ \mathrm{constant}


The answer is:

x2(x24x+1)+constantx^{2} \left(\frac{x^{2}}{4} - x + 1\right)+ \mathrm{constant}

The answer (Indefinite) [src]
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 | x*(x - 1)*(x - 2) dx = C + x  - x  + --
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x(x1)(x2)dx=C+x44x3+x2\int x \left(x - 1\right) \left(x - 2\right)\, dx = C + \frac{x^{4}}{4} - x^{3} + x^{2}
The graph
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The answer [src]
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14\frac{1}{4}
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14\frac{1}{4}
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Numerical answer [src]
0.25
0.25
The graph
Integral of x(x-1)(x-2) dx

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