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

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

Let $u = 1 - 2 \sin{\left(x \right)}$ .

Then let $du = - 2 \cos{\left(x \right)} dx$ and substitute $- \frac{du}{2}$ :

$\int \left(- \frac{\sqrt{u}}{2}\right)\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \sqrt{u}\, du = - \frac{\int \sqrt{u}\, du}{2}$
  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int \sqrt{u}\, du = \frac{2 u^{\frac{3}{2}}}{3}$
  So, the result is: $- \frac{u^{\frac{3}{2}}}{3}$
Now substitute $u$ back in:

$- \frac{\left(1 - 2 \sin{\left(x \right)}\right)^{\frac{3}{2}}}{3}$
Add the constant of integration:

$- \frac{\left(1 - 2 \sin{\left(x \right)}\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}$

The answer is:

$- \frac{\left(1 - 2 \sin{\left(x \right)}\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                  
 |                                                3/2
 |   ______________                 (1 - 2*sin(x))   
 | \/ 1 - 2*sin(x) *cos(x) dx = C - -----------------
 |                                          3        
/

$$\int \sqrt{1 - 2 \sin{\left(x \right)}} \cos{\left(x \right)}\, dx = C - \frac{\left(1 - 2 \sin{\left(x \right)}\right)^{\frac{3}{2}}}{3}$$

Simplify

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.

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

Limits of integration:

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Piecewise:

The solution