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Solving integrals/ 2x^2-x+1

Integral of 2x^2-x+1 dx

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The solution

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∫12((2x2−x)+1) dx\int\limits_{1}^{2} \left(\left(2 x^{2} - x\right) + 1\right)\, dx 
Integral(2*x^2 - x + 1, (x, 1, 2))
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:

        ∫2x2 dx=2∫x2 dx\int 2 x^{2}\, dx = 2 \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: 2x33\frac{2 x^{3}}{3}

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

        ∫(−x) dx=−∫x dx\int \left(- x\right)\, dx = - \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: −x22- \frac{x^{2}}{2}

      The result is: 2x33−x22\frac{2 x^{3}}{3} - \frac{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: 2x33−x22+x\frac{2 x^{3}}{3} - \frac{x^{2}}{2} + x

  2. Now simplify:

    x(4x2−3x+6)6\frac{x \left(4 x^{2} - 3 x + 6\right)}{6}

  3. Add the constant of integration:

    x(4x2−3x+6)6+constant\frac{x \left(4 x^{2} - 3 x + 6\right)}{6}+ \mathrm{constant}

The answer is:

x(4x2−3x+6)6+constant\frac{x \left(4 x^{2} - 3 x + 6\right)}{6}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                     
 |                              2      3
 | /   2        \              x    2*x 
 | \2*x  - x + 1/ dx = C + x - -- + ----
 |                             2     3  
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∫((2x2−x)+1) dx=C+2x33−x22+x\int \left(\left(2 x^{2} - x\right) + 1\right)\, dx = C + \frac{2 x^{3}}{3} - \frac{x^{2}}{2} + x
Simplify
The graph
1.002.001.101.201.301.401.501.601.701.801.90010
Plot the graph f(x)
The answer [src] 
25/6
256\frac{25}{6} 
=
= 
25/6
256\frac{25}{6} 
25/6
Numerical answer [src] 
4.16666666666667
4.16666666666667

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

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