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Solving integrals/ x^3-3x+1

Integral of x^3-3x+1 dx

Limits of integration:

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The graph:

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Piecewise:

The solution

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

  1. Integrate term-by-term:

    1. 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:

        ∫x3 dx=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:

        ∫(−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}

      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:

      ∫1 dx=x\int 1\, dx = x

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

  2. Now simplify:

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

  3. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
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∫((x3−3x)+1) dx=C+x44−3x22+x\int \left(\left(x^{3} - 3 x\right) + 1\right)\, dx = C + \frac{x^{4}}{4} - \frac{3 x^{2}}{2} + x
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.902-2
Plot the graph f(x)
The answer [src] 
-1/4
−14- \frac{1}{4} 
=
= 
-1/4
−14- \frac{1}{4} 
-1/4
Numerical answer [src] 
-0.25
-0.25

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

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