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Solving integrals/ -x^3+3x-2

Integral of -x^3+3x-2 dx

Limits of integration:

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Piecewise:

The solution

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∫01((−x3+3x)−2) dx\int\limits_{0}^{1} \left(\left(- x^{3} + 3 x\right) - 2\right)\, dx 
Integral(-x^3 + 3*x - 2, (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:

        ∫(−x3) dx=−∫x3 dx\int \left(- x^{3}\right)\, dx = - \int x^{3}\, dx

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

          ∫x3 dx=x44\int x^{3}\, dx = \frac{x^{4}}{4}

        So, the result is: −x44- \frac{x^{4}}{4}

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

        ∫3x dx=3∫x dx\int 3 x\, 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}

      The result is: −x44+3x22- \frac{x^{4}}{4} + \frac{3 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

    The result is: −x44+3x22−2x- \frac{x^{4}}{4} + \frac{3 x^{2}}{2} - 2 x

  2. Now simplify:

    x(−x3+6x−8)4\frac{x \left(- x^{3} + 6 x - 8\right)}{4}

  3. Add the constant of integration:

    x(−x3+6x−8)4+constant\frac{x \left(- x^{3} + 6 x - 8\right)}{4}+ \mathrm{constant}

The answer is:

x(−x3+6x−8)4+constant\frac{x \left(- x^{3} + 6 x - 8\right)}{4}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                         
 |                                  4      2
 | /   3          \                x    3*x 
 | \- x  + 3*x - 2/ dx = C - 2*x - -- + ----
 |                                 4     2  
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∫((−x3+3x)−2) dx=C−x44+3x22−2x\int \left(\left(- x^{3} + 3 x\right) - 2\right)\, dx = C - \frac{x^{4}}{4} + \frac{3 x^{2}}{2} - 2 x
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.902-4
Plot the graph f(x)
The answer [src] 
-3/4
−34- \frac{3}{4} 
=
= 
-3/4
−34- \frac{3}{4} 
-3/4
Numerical answer [src] 
-0.75
-0.75

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

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