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Solving integrals/ (x³+3x²-2x+3)

Integral of (x³+3x²-2x+3) dx

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The solution

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∫01((−2x+(x3+3x2))+3) dx\int\limits_{0}^{1} \left(\left(- 2 x + \left(x^{3} + 3 x^{2}\right)\right) + 3\right)\, dx 
Integral(x^3 + 3*x^2 - 2*x + 3, (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:

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

      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:

          ∫3x2 dx=3∫x2 dx\int 3 x^{2}\, dx = 3 \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: x3x^{3}

        The result is: x44+x3\frac{x^{4}}{4} + x^{3}

      The result is: x44+x3−x2\frac{x^{4}}{4} + x^{3} - x^{2}

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

      ∫3 dx=3x\int 3\, dx = 3 x

    The result is: x44+x3−x2+3x\frac{x^{4}}{4} + x^{3} - x^{2} + 3 x

  2. Now simplify:

    x(x34+x2−x+3)x \left(\frac{x^{3}}{4} + x^{2} - x + 3\right)

  3. Add the constant of integration:

    x(x34+x2−x+3)+constantx \left(\frac{x^{3}}{4} + x^{2} - x + 3\right)+ \mathrm{constant}

The answer is:

x(x34+x2−x+3)+constantx \left(\frac{x^{3}}{4} + x^{2} - x + 3\right)+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                 
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 | / 3      2          \           3    2         x 
 | \x  + 3*x  - 2*x + 3/ dx = C + x  - x  + 3*x + --
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∫((−2x+(x3+3x2))+3) dx=C+x44+x3−x2+3x\int \left(\left(- 2 x + \left(x^{3} + 3 x^{2}\right)\right) + 3\right)\, dx = C + \frac{x^{4}}{4} + x^{3} - x^{2} + 3 x
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90010
Plot the graph f(x)
The answer [src] 
13/4
134\frac{13}{4} 
=
= 
13/4
134\frac{13}{4} 
13/4
Numerical answer [src] 
3.25
3.25

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

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