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Solving integrals/ (sqrt(1-2sinx))*cosx

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

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The solution

You have entered [src] 
  p                           
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  6                           
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 |  \/ 1 - 2*sin(x) *cos(x) dx
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∫0p61−2sin⁡(x)cos⁡(x) dx\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 u=1−2sin⁡(x)u = 1 - 2 \sin{\left(x \right)}.

    Then let du=−2cos⁡(x)dxdu = - 2 \cos{\left(x \right)} dx and substitute −du2- \frac{du}{2}:

    ∫(−u2) du\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:

      ∫u du=−∫u du2\int \sqrt{u}\, du = - \frac{\int \sqrt{u}\, du}{2}

      1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

        ∫u du=2u323\int \sqrt{u}\, du = \frac{2 u^{\frac{3}{2}}}{3}

      So, the result is: −u323- \frac{u^{\frac{3}{2}}}{3}

    Now substitute uu back in:

    −(1−2sin⁡(x))323- \frac{\left(1 - 2 \sin{\left(x \right)}\right)^{\frac{3}{2}}}{3}

  2. Add the constant of integration:

    −(1−2sin⁡(x))323+constant- \frac{\left(1 - 2 \sin{\left(x \right)}\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}

The answer is:

−(1−2sin⁡(x))323+constant- \frac{\left(1 - 2 \sin{\left(x \right)}\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                  
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 |   ______________                 (1 - 2*sin(x))   
 | \/ 1 - 2*sin(x) *cos(x) dx = C - -----------------
 |                                          3        
/
∫1−2sin⁡(x)cos⁡(x) dx=C−(1−2sin⁡(x))323\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
21−2sin⁡(p6)sin⁡(p6)3−1−2sin⁡(p6)3+13\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
21−2sin⁡(p6)sin⁡(p6)3−1−2sin⁡(p6)3+13\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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