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Identical expressions
(three ÷(two ×sqrt(3x+ four))-x)
(3÷(2× square root of (3x plus 4)) minus x)
(three ÷(two × square root of (3x plus four)) minus x)
(3÷(2×√(3x+4))-x)
3÷2×sqrt3x+4-x
(3÷(2×sqrt(3x+4))-x)dx
Similar expressions
(3÷(2×sqrt(3x+4))+x)
(3÷(2×sqrt(3x-4))-x)

Integral of (3÷(2×sqrt(3x+4))-x) dx

The solution

You have entered [src]

 pi                       
 --                       
 2                        
  /                       
 |                        
 |  /      3          \   
 |  |------------- - x| dx
 |  |    _________    |   
 |  \2*\/ 3*x + 4     /   
 |                        
/                         
pi                        
--                        
4

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

Integral(3/((2*sqrt(3*x + 4))) - x, (x, pi/4, pi/2))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- x\right)\, dx = - \int x\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x\, dx = \frac{x^{2}}{2}$
  So, the result is: $- \frac{x^{2}}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{3}{2 \sqrt{3 x + 4}}\, dx = 3 \int \frac{1}{2 \sqrt{3 x + 4}}\, dx$
  1. Let $u = 2 \sqrt{3 x + 4}$ .
    
    Then let $du = \frac{3 dx}{\sqrt{3 x + 4}}$ and substitute $\frac{du}{6}$ :
    
    $\int \frac{1}{6}\, du$
    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: $\frac{u}{6}$
    Now substitute $u$ back in:
    
    $\frac{\sqrt{3 x + 4}}{3}$
  So, the result is: $\sqrt{3 x + 4}$
The result is: $- \frac{x^{2}}{2} + \sqrt{3 x + 4}$
Now simplify:

$- \frac{x^{2}}{2} + \sqrt{3 x + 4}$
Add the constant of integration:

$- \frac{x^{2}}{2} + \sqrt{3 x + 4}+ \mathrm{constant}$

The answer is:

$- \frac{x^{2}}{2} + \sqrt{3 x + 4}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

    __________       __________       2
   /     3*pi       /     3*pi    3*pi 
  /  4 + ----  -   /  4 + ----  - -----
\/        2      \/        4        32

$$- \sqrt{\frac{3 \pi}{4} + 4} - \frac{3 \pi^{2}}{32} + \sqrt{4 + \frac{3 \pi}{2}}$$

    __________       __________       2
   /     3*pi       /     3*pi    3*pi 
  /  4 + ----  -   /  4 + ----  - -----
\/        2      \/        4        32

$$- \sqrt{\frac{3 \pi}{4} + 4} - \frac{3 \pi^{2}}{32} + \sqrt{4 + \frac{3 \pi}{2}}$$

sqrt(4 + 3*pi/2) - sqrt(4 + 3*pi/4) - 3*pi^2/32

Numerical answer [src]

-0.494749228405482

-0.494749228405482

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

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Integral of (3÷(2×sqrt(3x+4))-x) dx

Limits of integration:

The graph:

Piecewise:

The solution