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Integral of (2*sqrt(x)-sqrt(2x)+5) dx

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The solution

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 |  \2*\/ x  - \/ 2*x  + 5/ dx
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01((2x2x)+5)dx\int\limits_{0}^{1} \left(\left(2 \sqrt{x} - \sqrt{2 x}\right) + 5\right)\, dx
Integral(2*sqrt(x) - sqrt(2*x) + 5, (x, 0, 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:

        2xdx=2xdx\int 2 \sqrt{x}\, dx = 2 \int \sqrt{x}\, dx

        1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n1n \neq -1:

          xdx=2x323\int \sqrt{x}\, dx = \frac{2 x^{\frac{3}{2}}}{3}

        So, the result is: 4x323\frac{4 x^{\frac{3}{2}}}{3}

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

        (2x)dx=2xdx\int \left(- \sqrt{2 x}\right)\, dx = - \int \sqrt{2 x}\, dx

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

          But the integral is

          22x323\frac{2 \sqrt{2} x^{\frac{3}{2}}}{3}

        So, the result is: 22x323- \frac{2 \sqrt{2} x^{\frac{3}{2}}}{3}

      The result is: 22x323+4x323- \frac{2 \sqrt{2} x^{\frac{3}{2}}}{3} + \frac{4 x^{\frac{3}{2}}}{3}

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

      5dx=5x\int 5\, dx = 5 x

    The result is: 22x323+4x323+5x- \frac{2 \sqrt{2} x^{\frac{3}{2}}}{3} + \frac{4 x^{\frac{3}{2}}}{3} + 5 x

  2. Add the constant of integration:

    22x323+4x323+5x+constant- \frac{2 \sqrt{2} x^{\frac{3}{2}}}{3} + \frac{4 x^{\frac{3}{2}}}{3} + 5 x+ \mathrm{constant}


The answer is:

22x323+4x323+5x+constant- \frac{2 \sqrt{2} x^{\frac{3}{2}}}{3} + \frac{4 x^{\frac{3}{2}}}{3} + 5 x+ \mathrm{constant}

The answer (Indefinite) [src]
  /                                                            
 |                                           3/2       ___  3/2
 | /    ___     _____    \                4*x      2*\/ 2 *x   
 | \2*\/ x  - \/ 2*x  + 5/ dx = C + 5*x + ------ - ------------
 |                                          3           3      
/                                                              
((2x2x)+5)dx=C22x323+4x323+5x\int \left(\left(2 \sqrt{x} - \sqrt{2 x}\right) + 5\right)\, dx = C - \frac{2 \sqrt{2} x^{\frac{3}{2}}}{3} + \frac{4 x^{\frac{3}{2}}}{3} + 5 x
The graph
0.001.000.100.200.300.400.500.600.700.800.90010
The answer [src]
         ___
19   2*\/ 2 
-- - -------
3       3   
193223\frac{19}{3} - \frac{2 \sqrt{2}}{3}
=
=
         ___
19   2*\/ 2 
-- - -------
3       3   
193223\frac{19}{3} - \frac{2 \sqrt{2}}{3}
19/3 - 2*sqrt(2)/3
Numerical answer [src]
5.39052429175127
5.39052429175127

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