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Integral of (sqrt(1-2sinx))*cosx dx

Limits of integration:

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The solution

You have entered [src]
  p                           
  -                           
  6                           
  /                           
 |                            
 |    ______________          
 |  \/ 1 - 2*sin(x) *cos(x) dx
 |                            
/                             
0                             
$$\int\limits_{0}^{\frac{p}{6}} \sqrt{1 - 2 \sin{\left(x \right)}} \cos{\left(x \right)}\, dx$$
Integral(sqrt(1 - 2*sin(x))*cos(x), (x, 0, p/6))
Detail solution
  1. Let .

    Then let and substitute :

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

      1. The integral of is when :

      So, the result is:

    Now substitute back in:

  2. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                  
 |                                                3/2
 |   ______________                 (1 - 2*sin(x))   
 | \/ 1 - 2*sin(x) *cos(x) dx = C - -----------------
 |                                          3        
/                                                    
$$\int \sqrt{1 - 2 \sin{\left(x \right)}} \cos{\left(x \right)}\, dx = C - \frac{\left(1 - 2 \sin{\left(x \right)}\right)^{\frac{3}{2}}}{3}$$
The answer [src]
        ______________         ______________       
       /          /p\         /          /p\     /p\
      /  1 - 2*sin|-|    2*  /  1 - 2*sin|-| *sin|-|
1   \/            \6/      \/            \6/     \6/
- - ------------------ + ---------------------------
3           3                         3             
$$\frac{2 \sqrt{1 - 2 \sin{\left(\frac{p}{6} \right)}} \sin{\left(\frac{p}{6} \right)}}{3} - \frac{\sqrt{1 - 2 \sin{\left(\frac{p}{6} \right)}}}{3} + \frac{1}{3}$$
=
=
        ______________         ______________       
       /          /p\         /          /p\     /p\
      /  1 - 2*sin|-|    2*  /  1 - 2*sin|-| *sin|-|
1   \/            \6/      \/            \6/     \6/
- - ------------------ + ---------------------------
3           3                         3             
$$\frac{2 \sqrt{1 - 2 \sin{\left(\frac{p}{6} \right)}} \sin{\left(\frac{p}{6} \right)}}{3} - \frac{\sqrt{1 - 2 \sin{\left(\frac{p}{6} \right)}}}{3} + \frac{1}{3}$$
1/3 - sqrt(1 - 2*sin(p/6))/3 + 2*sqrt(1 - 2*sin(p/6))*sin(p/6)/3

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