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Solving integrals/ x+2/x³+2x²+5xdx

Integral of x+2/x³+2x²+5xdx dx

Limits of integration:

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Piecewise:

The solution

You have entered [src] 
  1                         
  /                         
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 |  /    2       2      \   
 |  |x + -- + 2*x  + 5*x| dx
 |  |     3             |   
 |  \    x              /   
 |                          
/                           
0
∫01(5x+(2x2+(x+2x3))) dx\int\limits_{0}^{1} \left(5 x + \left(2 x^{2} + \left(x + \frac{2}{x^{3}}\right)\right)\right)\, dx 
Integral(x + 2/x^3 + 2*x^2 + 5*x, (x, 0, 1))
Detail solution

  1. Integrate term-by-term:

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

      ∫5x dx=5∫x dx\int 5 x\, 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}

    1. Integrate term-by-term:

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

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

      1. Integrate term-by-term:

        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}

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

          ∫2x3 dx=2∫1x3 dx\int \frac{2}{x^{3}}\, dx = 2 \int \frac{1}{x^{3}}\, dx

          1. Don't know the steps in finding this integral.

            But the integral is

            −12x2- \frac{1}{2 x^{2}}

          So, the result is: −1x2- \frac{1}{x^{2}}

        The result is: x22−1x2\frac{x^{2}}{2} - \frac{1}{x^{2}}

      The result is: 2x33+x22−1x2\frac{2 x^{3}}{3} + \frac{x^{2}}{2} - \frac{1}{x^{2}}

    The result is: 2x33+3x2−1x2\frac{2 x^{3}}{3} + 3 x^{2} - \frac{1}{x^{2}}

  2. Now simplify:

    x4(2x+9)−33x2\frac{x^{4} \left(2 x + 9\right) - 3}{3 x^{2}}

  3. Add the constant of integration:

    x4(2x+9)−33x2+constant\frac{x^{4} \left(2 x + 9\right) - 3}{3 x^{2}}+ \mathrm{constant}

The answer is:

x4(2x+9)−33x2+constant\frac{x^{4} \left(2 x + 9\right) - 3}{3 x^{2}}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                               
 |                                               3
 | /    2       2      \          1       2   2*x 
 | |x + -- + 2*x  + 5*x| dx = C - -- + 3*x  + ----
 | |     3             |           2           3  
 | \    x              /          x               
 |                                                
/
∫(5x+(2x2+(x+2x3))) dx=C+2x33+3x2−1x2\int \left(5 x + \left(2 x^{2} + \left(x + \frac{2}{x^{3}}\right)\right)\right)\, dx = C + \frac{2 x^{3}}{3} + 3 x^{2} - \frac{1}{x^{2}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90-20000000000002000000000000
Plot the graph f(x)
The answer [src] 
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∞\infty 
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∞\infty 
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Numerical answer [src] 
1.83073007580698e+38
1.83073007580698e+38

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

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