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Solving integrals/ -x^2+4x+2

Integral of -x^2+4x+2 dx

Limits of integration:

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The solution

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∫13((−x2+4x)+2) dx\int\limits_{1}^{3} \left(\left(- x^{2} + 4 x\right) + 2\right)\, dx 
Integral(-x^2 + 4*x + 2, (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:

        ∫4x dx=4∫x dx\int 4 x\, dx = 4 \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: 2x22 x^{2}

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

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

      ∫2 dx=2x\int 2\, dx = 2 x

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

  2. Now simplify:

    x(−x2+6x+6)3\frac{x \left(- x^{2} + 6 x + 6\right)}{3}

  3. Add the constant of integration:

    x(−x2+6x+6)3+constant\frac{x \left(- x^{2} + 6 x + 6\right)}{3}+ \mathrm{constant}

The answer is:

x(−x2+6x+6)3+constant\frac{x \left(- x^{2} + 6 x + 6\right)}{3}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                         
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 | /   2          \                   2   x 
 | \- x  + 4*x + 2/ dx = C + 2*x + 2*x  - --
 |                                        3 
/
∫((−x2+4x)+2) dx=C−x33+2x2+2x\int \left(\left(- x^{2} + 4 x\right) + 2\right)\, dx = C - \frac{x^{3}}{3} + 2 x^{2} + 2 x
Simplify
The graph
1.03.01.21.41.61.82.02.22.42.62.8020
Plot the graph f(x)
The answer [src] 
34/3
343\frac{34}{3} 
=
= 
34/3
343\frac{34}{3} 
34/3
Numerical answer [src] 
11.3333333333333
11.3333333333333

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

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