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6x³+x²-2x+1/2x-1
Solving integrals/ 6x³+x²-2x+1/2x-1

Integral of 6x³+x²-2x+1/2x-1 dx

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The solution

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 |  /   3    2         x    \   
 |  |6*x  + x  - 2*x + - - 1| dx
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∫01((x2+(−2x+(6x3+x2)))−1) dx\int\limits_{0}^{1} \left(\left(\frac{x}{2} + \left(- 2 x + \left(6 x^{3} + x^{2}\right)\right)\right) - 1\right)\, dx 
Integral(6*x^3 + x^2 - 2*x + x/2 - 1, (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:

        ∫x2 dx=∫x dx2\int \frac{x}{2}\, dx = \frac{\int x\, dx}{2}

        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: x24\frac{x^{2}}{4}

      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 a constant times a function is the constant times the integral of the function:

            ∫6x3 dx=6∫x3 dx\int 6 x^{3}\, dx = 6 \int x^{3}\, dx

            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}

            So, the result is: 3x42\frac{3 x^{4}}{2}

          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}

          The result is: 3x42+x33\frac{3 x^{4}}{2} + \frac{x^{3}}{3}

        The result is: 3x42+x33−x2\frac{3 x^{4}}{2} + \frac{x^{3}}{3} - x^{2}

      The result is: 3x42+x33−3x24\frac{3 x^{4}}{2} + \frac{x^{3}}{3} - \frac{3 x^{2}}{4}

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

      ∫(−1) dx=−x\int \left(-1\right)\, dx = - x

    The result is: 3x42+x33−3x24−x\frac{3 x^{4}}{2} + \frac{x^{3}}{3} - \frac{3 x^{2}}{4} - x

  2. Now simplify:

    x(18x3+4x2−9x−12)12\frac{x \left(18 x^{3} + 4 x^{2} - 9 x - 12\right)}{12}

  3. Add the constant of integration:

    x(18x3+4x2−9x−12)12+constant\frac{x \left(18 x^{3} + 4 x^{2} - 9 x - 12\right)}{12}+ \mathrm{constant}

The answer is:

x(18x3+4x2−9x−12)12+constant\frac{x \left(18 x^{3} + 4 x^{2} - 9 x - 12\right)}{12}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                       
 |                                           2    3      4
 | /   3    2         x    \              3*x    x    3*x 
 | |6*x  + x  - 2*x + - - 1| dx = C - x - ---- + -- + ----
 | \                  2    /               4     3     2  
 |                                                        
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∫((x2+(−2x+(6x3+x2)))−1) dx=C+3x42+x33−3x24−x\int \left(\left(\frac{x}{2} + \left(- 2 x + \left(6 x^{3} + x^{2}\right)\right)\right) - 1\right)\, dx = C + \frac{3 x^{4}}{2} + \frac{x^{3}}{3} - \frac{3 x^{2}}{4} - x
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.905-5
Plot the graph f(x)
The answer [src] 
1/12
112\frac{1}{12} 
=
= 
1/12
112\frac{1}{12} 
1/12
Numerical answer [src] 
0.0833333333333333
0.0833333333333333
The graph
Integral of 6x³+x²-2x+1/2x-1 dx

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

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