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Solving integrals/ (2*root(x)-x^2+3)/root(3,x)

Integral of (2*root(x)-x^2+3)/root(3,x) dx

The solution

You have entered [src]

  1                    
  /                    
 |                     
 |      ___    2       
 |  2*\/ x  - x  + 3   
 |  ---------------- dx
 |         ___         
 |       \/ 3          
 |                     
/                      
0

$$\int\limits_{0}^{1} \frac{\left(2 \sqrt{x} - x^{2}\right) + 3}{\sqrt{3}}\, dx$$

Integral((2*sqrt(x) - x^2 + 3)/sqrt(3), (x, 0, 1))

Detail solution

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

$\int \frac{\left(2 \sqrt{x} - x^{2}\right) + 3}{\sqrt{3}}\, dx = \frac{\sqrt{3}}{3} \int \left(\left(2 \sqrt{x} - x^{2}\right) + 3\right)\, dx$
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:
      
      $\int 2 \sqrt{x}\, dx = 2 \int \sqrt{x}\, dx$
      1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
        
        $\int \sqrt{x}\, dx = \frac{2 x^{\frac{3}{2}}}{3}$
      So, the result is: $\frac{4 x^{\frac{3}{2}}}{3}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- x^{2}\right)\, dx = - \int x^{2}\, dx$
      1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
        
        $\int x^{2}\, dx = \frac{x^{3}}{3}$
      So, the result is: $- \frac{x^{3}}{3}$
    The result is: $\frac{4 x^{\frac{3}{2}}}{3} - \frac{x^{3}}{3}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int 3\, dx = 3 x$
  The result is: $\frac{4 x^{\frac{3}{2}}}{3} - \frac{x^{3}}{3} + 3 x$
So, the result is: $\frac{\sqrt{3}}{3} \left(\frac{4 x^{\frac{3}{2}}}{3} - \frac{x^{3}}{3} + 3 x\right)$
Now simplify:

$\frac{\sqrt{3} \left(4 x^{\frac{3}{2}} - x^{3} + 9 x\right)}{9}$
Add the constant of integration:

$\frac{\sqrt{3} \left(4 x^{\frac{3}{2}} - x^{3} + 9 x\right)}{9}+ \mathrm{constant}$

The answer is:

$\frac{\sqrt{3} \left(4 x^{\frac{3}{2}} - x^{3} + 9 x\right)}{9}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                   
 |                                                    
 |     ___    2                ___ /       3      3/2\
 | 2*\/ x  - x  + 3          \/ 3  |      x    4*x   |
 | ---------------- dx = C + -----*|3*x - -- + ------|
 |        ___                  3   \      3      3   /
 |      \/ 3                                          
 |                                                    
/

$$\int \frac{\left(2 \sqrt{x} - x^{2}\right) + 3}{\sqrt{3}}\, dx = C + \frac{\sqrt{3}}{3} \left(\frac{4 x^{\frac{3}{2}}}{3} - \frac{x^{3}}{3} + 3 x\right)$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

    ___
4*\/ 3 
-------
   3

$$\frac{4 \sqrt{3}}{3}$$

    ___
4*\/ 3 
-------
   3

$$\frac{4 \sqrt{3}}{3}$$

4*sqrt(3)/3

Numerical answer [src]

2.3094010767585

2.3094010767585

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

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Integral of (2*root(x)-x^2+3)/root(3,x) dx

Limits of integration:

The graph:

Piecewise:

The solution