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Solving integrals/ sqrt(2+2*sin(x))

Integral of sqrt(2+2*sin(x)) dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

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∫−π202sin⁡(x)+2 dx\int\limits_{- \frac{\pi}{2}}^{0} \sqrt{2 \sin{\left(x \right)} + 2}\, dx 
Integral(sqrt(2 + 2*sin(x)), (x, -pi/2, 0))
Detail solution

  1. Rewrite the integrand:

    2sin⁡(x)+2=2sin⁡(x)+1\sqrt{2 \sin{\left(x \right)} + 2} = \sqrt{2} \sqrt{\sin{\left(x \right)} + 1}

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

    ∫2sin⁡(x)+1 dx=2∫sin⁡(x)+1 dx\int \sqrt{2} \sqrt{\sin{\left(x \right)} + 1}\, dx = \sqrt{2} \int \sqrt{\sin{\left(x \right)} + 1}\, dx

    1. Don't know the steps in finding this integral.

      But the integral is

      ∫sin⁡(x)+1 dx\int \sqrt{\sin{\left(x \right)} + 1}\, dx

    So, the result is: 2∫sin⁡(x)+1 dx\sqrt{2} \int \sqrt{\sin{\left(x \right)} + 1}\, dx

  3. Add the constant of integration:

    2∫sin⁡(x)+1 dx+constant\sqrt{2} \int \sqrt{\sin{\left(x \right)} + 1}\, dx+ \mathrm{constant}

The answer is:

2∫sin⁡(x)+1 dx+constant\sqrt{2} \int \sqrt{\sin{\left(x \right)} + 1}\, dx+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                  /                 
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 |   ______________            ___  |   ____________   
 | \/ 2 + 2*sin(x)  dx = C + \/ 2 * | \/ 1 + sin(x)  dx
 |                                  |                  
/                                  /
∫2sin⁡(x)+2 dx=C+2∫sin⁡(x)+1 dx\int \sqrt{2 \sin{\left(x \right)} + 2}\, dx = C + \sqrt{2} \int \sqrt{\sin{\left(x \right)} + 1}\, dx
Simplify
Numerical answer [src] 
1.17157287525381
1.17157287525381

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

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