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32x(2x-1)(4x^2-16x+15)
Solving integrals/ 32x(2x-1)(4x^2-16x+15)

Integral of 32x(2x-1)(4x^2-16x+15) dx

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The solution

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∫0132x(2x−1)(4x2−16x+15) dx\int\limits_{0}^{1} 32 x \left(2 x - 1\right) \left(4 x^{2} - 16 x + 15\right)\, dx
Simplify
Detail solution

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

    ∫32x(2x−1)(4x2−16x+15) dx=32∫x(2x−1)(4x2−16x+15) dx\int 32 x \left(2 x - 1\right) \left(4 x^{2} - 16 x + 15\right)\, dx = 32 \int x \left(2 x - 1\right) \left(4 x^{2} - 16 x + 15\right)\, dx

    1. Rewrite the integrand:

      x(2x−1)(4x2−16x+15)=8x4−36x3+46x2−15xx \left(2 x - 1\right) \left(4 x^{2} - 16 x + 15\right) = 8 x^{4} - 36 x^{3} + 46 x^{2} - 15 x

    2. Integrate term-by-term:

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

        ∫8x4 dx=8∫x4 dx\int 8 x^{4}\, dx = 8 \int x^{4}\, dx

        1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

          ∫x4 dx=x55\int x^{4}\, dx = \frac{x^{5}}{5}

        So, the result is: 8x55\frac{8 x^{5}}{5}

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

        ∫(−36x3) dx=−36∫x3 dx\int \left(- 36 x^{3}\right)\, dx = - 36 \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: −9x4- 9 x^{4}

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

        ∫46x2 dx=46∫x2 dx\int 46 x^{2}\, dx = 46 \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: 46x33\frac{46 x^{3}}{3}

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

        ∫(−15x) dx=−15∫x dx\int \left(- 15 x\right)\, dx = - 15 \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: −15x22- \frac{15 x^{2}}{2}

      The result is: 8x55−9x4+46x33−15x22\frac{8 x^{5}}{5} - 9 x^{4} + \frac{46 x^{3}}{3} - \frac{15 x^{2}}{2}

    So, the result is: 256x55−288x4+1472x33−240x2\frac{256 x^{5}}{5} - 288 x^{4} + \frac{1472 x^{3}}{3} - 240 x^{2}

  2. Now simplify:

    16x2⋅(48x3−270x2+460x−225)15\frac{16 x^{2} \cdot \left(48 x^{3} - 270 x^{2} + 460 x - 225\right)}{15}

  3. Add the constant of integration:

    16x2⋅(48x3−270x2+460x−225)15+constant\frac{16 x^{2} \cdot \left(48 x^{3} - 270 x^{2} + 460 x - 225\right)}{15}+ \mathrm{constant}

The answer is:

16x2⋅(48x3−270x2+460x−225)15+constant\frac{16 x^{2} \cdot \left(48 x^{3} - 270 x^{2} + 460 x - 225\right)}{15}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                                             
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 |                /   2            \               4        2   256*x    1472*x 
 | 32*x*(2*x - 1)*\4*x  - 16*x + 15/ dx = C - 288*x  - 240*x  + ------ + -------
 |                                                                5         3   
/
16 (48 x5−270 x4+460 x3−225 x2)15{{16\,\left(48\,x^5-270\,x^4+460\,x^3-225\,x^2\right)}\over{15}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90-200200
Plot the graph f(x)
The answer [src] 
208
---
 15
20815{{208}\over{15}} 
=
= 
208
---
 15
20815\frac{208}{15}
Simplify
Numerical answer [src] 
13.8666666666667
13.8666666666667
The graph
Integral of 32x(2x-1)(4x^2-16x+15) dx

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

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