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Identical expressions
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Similar expressions
2cosx+2sqrt(3)sinx

Integral of 2cosx-2sqrt(3)sinx dx

The solution

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 pi                               
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 6                                
  /                               
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 |  /               ___       \   
 |  \2*cos(x) - 2*\/ 3 *sin(x)/ dx
 |                                
/                                 
0

$$\int\limits_{0}^{\frac{\pi}{6}} \left(- 2 \sqrt{3} \sin{\left(x \right)} + 2 \cos{\left(x \right)}\right)\, dx$$

Integral(2*cos(x) - 2*sqrt(3)*sin(x), (x, 0, pi/6))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 2 \sqrt{3} \sin{\left(x \right)}\right)\, dx = - 2 \sqrt{3} \int \sin{\left(x \right)}\, dx$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
  So, the result is: $2 \sqrt{3} \cos{\left(x \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 2 \cos{\left(x \right)}\, dx = 2 \int \cos{\left(x \right)}\, dx$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
  So, the result is: $2 \sin{\left(x \right)}$
The result is: $2 \sin{\left(x \right)} + 2 \sqrt{3} \cos{\left(x \right)}$
Now simplify:

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

$4 \sin{\left(x + \frac{\pi}{3} \right)}+ \mathrm{constant}$

The answer is:

$4 \sin{\left(x + \frac{\pi}{3} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                              
 |                                                               
 | /               ___       \                         ___       
 | \2*cos(x) - 2*\/ 3 *sin(x)/ dx = C + 2*sin(x) + 2*\/ 3 *cos(x)
 |                                                               
/

$$\int \left(- 2 \sqrt{3} \sin{\left(x \right)} + 2 \cos{\left(x \right)}\right)\, dx = C + 2 \sin{\left(x \right)} + 2 \sqrt{3} \cos{\left(x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

        ___
4 - 2*\/ 3

$$4 - 2 \sqrt{3}$$

        ___
4 - 2*\/ 3

$$4 - 2 \sqrt{3}$$

4 - 2*sqrt(3)

Numerical answer [src]

0.535898384862245

0.535898384862245

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

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Integral of 2cosx-2sqrt(3)sinx dx

Limits of integration:

The graph:

Piecewise:

The solution