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Solving integrals/ -x²+2x+3

Integral of -x²+2x+3 dx

Limits of integration:

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Piecewise:

The solution

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∫−13((−x2+2x)+3) dx\int\limits_{-1}^{3} \left(\left(- x^{2} + 2 x\right) + 3\right)\, dx 
Integral(-x^2 + 2*x + 3, (x, -1, 3))
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:

        ∫(−x2) dx=−∫x2 dx\int \left(- x^{2}\right)\, dx = - \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: −x33- \frac{x^{3}}{3}

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

        ∫2x dx=2∫x dx\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 n≠−1n \neq -1:

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

        So, the result is: x2x^{2}

      The result is: −x33+x2- \frac{x^{3}}{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: −x33+x2+3x- \frac{x^{3}}{3} + x^{2} + 3 x

  2. Now simplify:

    x(−x2+3x+9)3\frac{x \left(- x^{2} + 3 x + 9\right)}{3}

  3. Add the constant of integration:

    x(−x2+3x+9)3+constant\frac{x \left(- x^{2} + 3 x + 9\right)}{3}+ \mathrm{constant}

The answer is:

x(−x2+3x+9)3+constant\frac{x \left(- x^{2} + 3 x + 9\right)}{3}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                       
 |                                       3
 | /   2          \           2         x 
 | \- x  + 2*x + 3/ dx = C + x  + 3*x - --
 |                                      3 
/
∫((−x2+2x)+3) dx=C−x33+x2+3x\int \left(\left(- x^{2} + 2 x\right) + 3\right)\, dx = C - \frac{x^{3}}{3} + x^{2} + 3 x
Simplify
The graph
-1.0-0.53.00.00.51.01.52.02.5-1010
Plot the graph f(x)
The answer [src] 
32/3
323\frac{32}{3} 
=
= 
32/3
323\frac{32}{3} 
32/3
Numerical answer [src] 
10.6666666666667
10.6666666666667

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

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