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Similar expressions
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Solving integrals/ (4sqrt(x)+2/(sqrt(x)))

Integral of (4sqrt(x)+2/(sqrt(x))) dx

The solution

You have entered [src]

  9                     
  /                     
 |                      
 |  /    ___     2  \   
 |  |4*\/ x  + -----| dx
 |  |            ___|   
 |  \          \/ x /   
 |                      
/                       
4

$$\int\limits_{4}^{9} \left(4 \sqrt{x} + \frac{2}{\sqrt{x}}\right)\, dx$$

Integral(4*sqrt(x) + 2/sqrt(x), (x, 4, 9))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 4 \sqrt{x}\, dx = 4 \int \sqrt{x}\, dx$
  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: $\frac{8 x^{\frac{3}{2}}}{3}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{2}{\sqrt{x}}\, dx = 2 \int \frac{1}{\sqrt{x}}\, dx$
  1. Let $u = \sqrt{x}$ .
    
    Then let $du = \frac{dx}{2 \sqrt{x}}$ and substitute $2 du$ :
    
    $\int 2\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\text{False}$
      1. The integral of a constant is the constant times the variable of integration:
        
        $\int 1\, du = u$
      So, the result is: $2 u$
    Now substitute $u$ back in:
    
    $2 \sqrt{x}$
  So, the result is: $4 \sqrt{x}$
The result is: $\frac{8 x^{\frac{3}{2}}}{3} + 4 \sqrt{x}$
Now simplify:

$\sqrt{x} \left(\frac{8 x}{3} + 4\right)$
Add the constant of integration:

$\sqrt{x} \left(\frac{8 x}{3} + 4\right)+ \mathrm{constant}$

The answer is:

$\sqrt{x} \left(\frac{8 x}{3} + 4\right)+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                           
 |                                         3/2
 | /    ___     2  \              ___   8*x   
 | |4*\/ x  + -----| dx = C + 4*\/ x  + ------
 | |            ___|                      3   
 | \          \/ x /                          
 |                                            
/

$$\int \left(4 \sqrt{x} + \frac{2}{\sqrt{x}}\right)\, dx = C + \frac{8 x^{\frac{3}{2}}}{3} + 4 \sqrt{x}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

164/3

$$\frac{164}{3}$$

164/3

$$\frac{164}{3}$$

164/3

Numerical answer [src]

54.6666666666667

54.6666666666667

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

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Integral of (4sqrt(x)+2/(sqrt(x))) dx

Limits of integration:

The graph:

Piecewise:

The solution