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four *sqrt(two -2sin(x))
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4*sqrt(2+2sin(x))
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Integral of 4*sqrt(2-2sin(x)) dx

The solution

You have entered [src]

 pi                      
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 6                       
  /                      
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 |      ______________   
 |  4*\/ 2 - 2*sin(x)  dx
 |                       
/                        
0

$$\int\limits_{0}^{\frac{\pi}{6}} 4 \sqrt{2 - 2 \sin{\left(x \right)}}\, dx$$

Integral(4*sqrt(2 - 2*sin(x)), (x, 0, pi/6))

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int 4 \sqrt{2 - 2 \sin{\left(x \right)}}\, dx = 4 \int \sqrt{2 - 2 \sin{\left(x \right)}}\, dx$
1. Rewrite the integrand:
  
  $\sqrt{2 - 2 \sin{\left(x \right)}} = \sqrt{2} \sqrt{1 - \sin{\left(x \right)}}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \sqrt{2} \sqrt{1 - \sin{\left(x \right)}}\, dx = \sqrt{2} \int \sqrt{1 - \sin{\left(x \right)}}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\int \sqrt{1 - \sin{\left(x \right)}}\, dx$
  So, the result is: $\sqrt{2} \int \sqrt{1 - \sin{\left(x \right)}}\, dx$
So, the result is: $4 \sqrt{2} \int \sqrt{1 - \sin{\left(x \right)}}\, dx$
Add the constant of integration:

$4 \sqrt{2} \int \sqrt{1 - \sin{\left(x \right)}}\, dx+ \mathrm{constant}$

The answer is:

$4 \sqrt{2} \int \sqrt{1 - \sin{\left(x \right)}}\, dx+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                      /                 
 |                                      |                  
 |     ______________              ___  |   ____________   
 | 4*\/ 2 - 2*sin(x)  dx = C + 4*\/ 2 * | \/ 1 - sin(x)  dx
 |                                      |                  
/                                      /

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

Simplify

Numerical answer [src]

2.54269796156626

2.54269796156626

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

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Integral of 4*sqrt(2-2sin(x)) dx

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The solution