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Solving integrals/ (3x^2-4x-(2/x))

Integral of (3x^2-4x-(2/x)) dx

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Piecewise:

The solution

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∫21((3x2−4x)−2x) dx\int\limits_{2}^{1} \left(\left(3 x^{2} - 4 x\right) - \frac{2}{x}\right)\, dx 
Integral(3*x^2 - 4*x - 2/x, (x, 2, 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:

        ∫3x2 dx=3∫x2 dx\int 3 x^{2}\, dx = 3 \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: x3x^{3}

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

        ∫(−4x) dx=−4∫x dx\int \left(- 4 x\right)\, dx = - 4 \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: −2x2- 2 x^{2}

      The result is: x3−2x2x^{3} - 2 x^{2}

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

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

      1. The integral of 1x\frac{1}{x} is log⁡(x)\log{\left(x \right)}.

      So, the result is: −2log⁡(x)- 2 \log{\left(x \right)}

    The result is: x3−2x2−2log⁡(x)x^{3} - 2 x^{2} - 2 \log{\left(x \right)}

  2. Add the constant of integration:

    x3−2x2−2log⁡(x)+constantx^{3} - 2 x^{2} - 2 \log{\left(x \right)}+ \mathrm{constant}

The answer is:

x3−2x2−2log⁡(x)+constantx^{3} - 2 x^{2} - 2 \log{\left(x \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                              
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 | /   2         2\           3      2           
 | |3*x  - 4*x - -| dx = C + x  - 2*x  - 2*log(x)
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∫((3x2−4x)−2x) dx=C+x3−2x2−2log⁡(x)\int \left(\left(3 x^{2} - 4 x\right) - \frac{2}{x}\right)\, dx = C + x^{3} - 2 x^{2} - 2 \log{\left(x \right)}
Simplify
The graph
1.002.001.101.201.301.401.501.601.701.801.905-5
Plot the graph f(x)
The answer [src] 
-1 + 2*log(2)
−1+2log⁡(2)-1 + 2 \log{\left(2 \right)} 
=
= 
-1 + 2*log(2)
−1+2log⁡(2)-1 + 2 \log{\left(2 \right)} 
-1 + 2*log(2)
Numerical answer [src] 
0.386294361119891
0.386294361119891

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

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