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Solving integrals/ dx/(sqrt(3)*sqrt(x)+4)

Integral of dx/(sqrt(3)*sqrt(x)+4) dx

The solution

You have entered [src]

  4                   
  /                   
 |                    
 |         1          
 |  --------------- dx
 |    ___   ___       
 |  \/ 3 *\/ x  + 4   
 |                    
/                     
-1

$$\int\limits_{-1}^{4} \frac{1}{\sqrt{3} \sqrt{x} + 4}\, dx$$

Integral(1/(sqrt(3)*sqrt(x) + 4), (x, -1, 4))

Detail solution

Let $u = \sqrt{x}$ .

Then let $du = \frac{dx}{2 \sqrt{x}}$ and substitute $2 du$ :

$\int \frac{2 u}{\sqrt{3} u + 4}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{u}{\sqrt{3} u + 4}\, du = 2 \int \frac{u}{\sqrt{3} u + 4}\, du$
  1. Rewrite the integrand:
    
    $\frac{u}{\sqrt{3} u + 4} = \frac{\sqrt{3}}{3} - \frac{4 \sqrt{3}}{3 \left(\sqrt{3} u + 4\right)}$
  2. Integrate term-by-term:
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int \frac{\sqrt{3}}{3}\, du = \frac{\sqrt{3} u}{3}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{4 \sqrt{3}}{3 \left(\sqrt{3} u + 4\right)}\right)\, du = - \frac{4 \sqrt{3} \int \frac{1}{\sqrt{3} u + 4}\, du}{3}$
      1. Let $u = \sqrt{3} u + 4$ .
        
        Then let $du = \sqrt{3} du$ and substitute $\frac{\sqrt{3} du}{3}$ :
        
        $\int \frac{\sqrt{3}}{3 u}\, du$
        
        The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \frac{1}{u}\, du = \frac{\sqrt{3} \int \frac{1}{u}\, du}{3}$
        
        The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
        
        So, the result is: $\frac{\sqrt{3} \log{\left(u \right)}}{3}$
        
        Now substitute $u$ back in:
        
        $\frac{\sqrt{3} \log{\left(\sqrt{3} u + 4 \right)}}{3}$
      So, the result is: $- \frac{4 \log{\left(\sqrt{3} u + 4 \right)}}{3}$
    The result is: $\frac{\sqrt{3} u}{3} - \frac{4 \log{\left(\sqrt{3} u + 4 \right)}}{3}$
  So, the result is: $\frac{2 \sqrt{3} u}{3} - \frac{8 \log{\left(\sqrt{3} u + 4 \right)}}{3}$
Now substitute $u$ back in:

$\frac{2 \sqrt{3} \sqrt{x}}{3} - \frac{8 \log{\left(\sqrt{3} \sqrt{x} + 4 \right)}}{3}$
Add the constant of integration:

$\frac{2 \sqrt{3} \sqrt{x}}{3} - \frac{8 \log{\left(\sqrt{3} \sqrt{x} + 4 \right)}}{3}+ \mathrm{constant}$

The answer is:

$\frac{2 \sqrt{3} \sqrt{x}}{3} - \frac{8 \log{\left(\sqrt{3} \sqrt{x} + 4 \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                               
 |                               /      ___   ___\       ___   ___
 |        1                 8*log\4 + \/ 3 *\/ x /   2*\/ 3 *\/ x 
 | --------------- dx = C - ---------------------- + -------------
 |   ___   ___                        3                    3      
 | \/ 3 *\/ x  + 4                                                
 |                                                                
/

$$\int \frac{1}{\sqrt{3} \sqrt{x} + 4}\, dx = C + \frac{2 \sqrt{3} \sqrt{x}}{3} - \frac{8 \log{\left(\sqrt{3} \sqrt{x} + 4 \right)}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

       /        ___\       ___        /        ___\         ___
  8*log\4 + 2*\/ 3 /   4*\/ 3    8*log\4 + I*\/ 3 /   2*I*\/ 3 
- ------------------ + ------- + ------------------ - ---------
          3               3              3                3

$$- \frac{8 \log{\left(2 \sqrt{3} + 4 \right)}}{3} + \frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3} i}{3} + \frac{8 \log{\left(4 + \sqrt{3} i \right)}}{3}$$

       /        ___\       ___        /        ___\         ___
  8*log\4 + 2*\/ 3 /   4*\/ 3    8*log\4 + I*\/ 3 /   2*I*\/ 3 
- ------------------ + ------- + ------------------ - ---------
          3               3              3                3

$$- \frac{8 \log{\left(2 \sqrt{3} + 4 \right)}}{3} + \frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3} i}{3} + \frac{8 \log{\left(4 + \sqrt{3} i \right)}}{3}$$

-8*log(4 + 2*sqrt(3))/3 + 4*sqrt(3)/3 + 8*log(4 + i*sqrt(3))/3 - 2*i*sqrt(3)/3

Numerical answer [src]

(0.87495726530422 - 0.0650215499529942j)

(0.87495726530422 - 0.0650215499529942j)

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

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Integral of dx/(sqrt(3)*sqrt(x)+4) dx

Limits of integration:

The graph:

Piecewise:

The solution