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dx/(one +sqrt(three)*x)+ two
dx divide by (1 plus square root of (3) multiply by x) plus 2
dx divide by (one plus square root of (three) multiply by x) plus two
dx/(1+√(3)*x)+2
dx/(1+sqrt(3)x)+2
dx/1+sqrt3x+2
dx divide by (1+sqrt(3)*x)+2
Similar expressions
dx/(1-sqrt(3)*x)+2
dx/(1+sqrt(3)*x)-2

Integral of dx/(1+sqrt(3)*x)+2 dx

The solution

You have entered [src]

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

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

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

Detail solution

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

$2 x + \frac{\sqrt{3} \log{\left(\sqrt{3} x + 1 \right)}}{3}$
Add the constant of integration:

$2 x + \frac{\sqrt{3} \log{\left(\sqrt{3} x + 1 \right)}}{3}+ \mathrm{constant}$

The answer is:

$2 x + \frac{\sqrt{3} \log{\left(\sqrt{3} x + 1 \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

      ___    /      ___\
    \/ 3 *log\1 + \/ 3 /
2 + --------------------
             3

$$\frac{\sqrt{3} \log{\left(1 + \sqrt{3} \right)}}{3} + 2$$

      ___    /      ___\
    \/ 3 *log\1 + \/ 3 /
2 + --------------------
             3

$$\frac{\sqrt{3} \log{\left(1 + \sqrt{3} \right)}}{3} + 2$$

2 + sqrt(3)*log(1 + sqrt(3))/3

Numerical answer [src]

2.58026735379263

2.58026735379263

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

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Integral of dx/(1+sqrt(3)*x)+2 dx

Limits of integration:

The graph:

Piecewise:

The solution