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Solving integrals/ (7x-5)*x-2dx*(1/3)

Integral of (7x-5)*x-2dx*(1/3) dx

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The solution

You have entered [src] 
  1                          
  /                          
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 |  /              2*(-1)\   
 |  |(7*x - 5)*x + ------| dx
 |  \                3   /   
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∫31(x(7x−5)+(−1)23) dx\int\limits_{3}^{1} \left(x \left(7 x - 5\right) + \frac{\left(-1\right) 2}{3}\right)\, dx 
Integral((7*x - 5)*x + 2*(-1)/3, (x, 3, 1))
Detail solution

  1. Integrate term-by-term:

    1. Rewrite the integrand:

      x(7x−5)=7x2−5xx \left(7 x - 5\right) = 7 x^{2} - 5 x

    2. Integrate term-by-term:

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

        ∫7x2 dx=7∫x2 dx\int 7 x^{2}\, dx = 7 \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: 7x33\frac{7 x^{3}}{3}

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

        ∫(−5x) dx=−5∫x dx\int \left(- 5 x\right)\, dx = - 5 \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: −5x22- \frac{5 x^{2}}{2}

      The result is: 7x33−5x22\frac{7 x^{3}}{3} - \frac{5 x^{2}}{2}

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

      ∫(−1)23 dx=−2x3\int \frac{\left(-1\right) 2}{3}\, dx = - \frac{2 x}{3}

    The result is: 7x33−5x22−2x3\frac{7 x^{3}}{3} - \frac{5 x^{2}}{2} - \frac{2 x}{3}

  2. Now simplify:

    x(14x2−15x−4)6\frac{x \left(14 x^{2} - 15 x - 4\right)}{6}

  3. Add the constant of integration:

    x(14x2−15x−4)6+constant\frac{x \left(14 x^{2} - 15 x - 4\right)}{6}+ \mathrm{constant}

The answer is:

x(14x2−15x−4)6+constant\frac{x \left(14 x^{2} - 15 x - 4\right)}{6}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                 
 |                                    2            3
 | /              2*(-1)\          5*x    2*x   7*x 
 | |(7*x - 5)*x + ------| dx = C - ---- - --- + ----
 | \                3   /           2      3     3  
 |                                                  
/
∫(x(7x−5)+(−1)23) dx=C+7x33−5x22−2x3\int \left(x \left(7 x - 5\right) + \frac{\left(-1\right) 2}{3}\right)\, dx = C + \frac{7 x^{3}}{3} - \frac{5 x^{2}}{2} - \frac{2 x}{3}
Simplify
The graph
1.03.01.21.41.61.82.02.22.42.62.8-5050
Plot the graph f(x)
The answer [src] 
-118/3
−1183- \frac{118}{3} 
=
= 
-118/3
−1183- \frac{118}{3} 
-118/3
Numerical answer [src] 
-39.3333333333333
-39.3333333333333

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

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