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Solving integrals/ x*3*(x^2)-xdx

Integral of x*3*(x^2)-xdx dx

Limits of integration:

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The solution

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∫12(x2⋅3x−x) dx\int\limits_{1}^{2} \left(x^{2} \cdot 3 x - x\right)\, dx 
Integral((x*3)*x^2 - x, (x, 1, 2))
Detail solution

  1. Integrate term-by-term:

    1. Let u=x2u = x^{2}.

      Then let du=2xdxdu = 2 x dx and substitute 3du2\frac{3 du}{2}:

      ∫3u2 du\int \frac{3 u}{2}\, du

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

        ∫u du=3∫u du2\int u\, du = \frac{3 \int u\, du}{2}

        1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

          ∫u du=u22\int u\, du = \frac{u^{2}}{2}

        So, the result is: 3u24\frac{3 u^{2}}{4}

      Now substitute uu back in:

      3x44\frac{3 x^{4}}{4}

    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: 3x44−x22\frac{3 x^{4}}{4} - \frac{x^{2}}{2}

  2. Now simplify:

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

  3. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                               
 |                        2      4
 | /     2    \          x    3*x 
 | \x*3*x  - x/ dx = C - -- + ----
 |                       2     4  
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∫(x2⋅3x−x) dx=C+3x44−x22\int \left(x^{2} \cdot 3 x - x\right)\, dx = C + \frac{3 x^{4}}{4} - \frac{x^{2}}{2}
Simplify
The graph
1.002.001.101.201.301.401.501.601.701.801.90025
Plot the graph f(x)
The answer [src] 
39/4
394\frac{39}{4} 
=
= 
39/4
394\frac{39}{4} 
39/4
Numerical answer [src] 
9.75
9.75

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

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