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Integral of x-3/2x^2+5x-7 dx

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The solution

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01((5x+(3x22+x))7)dx\int\limits_{0}^{1} \left(\left(5 x + \left(- \frac{3 x^{2}}{2} + x\right)\right) - 7\right)\, dx
Integral(x - 3*x^2/2 + 5*x - 7, (x, 0, 1))
Detail solution
  1. Integrate term-by-term:

    1. Integrate term-by-term:

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

        5xdx=5xdx\int 5 x\, dx = 5 \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: 5x22\frac{5 x^{2}}{2}

      1. Integrate term-by-term:

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

          (3x22)dx=3x2dx2\int \left(- \frac{3 x^{2}}{2}\right)\, dx = - \frac{3 \int x^{2}\, dx}{2}

          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: x32- \frac{x^{3}}{2}

        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}

        The result is: x32+x22- \frac{x^{3}}{2} + \frac{x^{2}}{2}

      The result is: x32+3x2- \frac{x^{3}}{2} + 3 x^{2}

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

      (7)dx=7x\int \left(-7\right)\, dx = - 7 x

    The result is: x32+3x27x- \frac{x^{3}}{2} + 3 x^{2} - 7 x

  2. Now simplify:

    x(x2+6x14)2\frac{x \left(- x^{2} + 6 x - 14\right)}{2}

  3. Add the constant of integration:

    x(x2+6x14)2+constant\frac{x \left(- x^{2} + 6 x - 14\right)}{2}+ \mathrm{constant}


The answer is:

x(x2+6x14)2+constant\frac{x \left(- x^{2} + 6 x - 14\right)}{2}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                                             
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 | |    3*x           |                   2   x 
 | |x - ---- + 5*x - 7| dx = C - 7*x + 3*x  - --
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((5x+(3x22+x))7)dx=Cx32+3x27x\int \left(\left(5 x + \left(- \frac{3 x^{2}}{2} + x\right)\right) - 7\right)\, dx = C - \frac{x^{3}}{2} + 3 x^{2} - 7 x
The graph
0.001.000.100.200.300.400.500.600.700.800.90-1010
The answer [src]
-9/2
92- \frac{9}{2}
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-9/2
92- \frac{9}{2}
-9/2
Numerical answer [src]
-4.5
-4.5

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