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Solving integrals/ sqrt(2sin(x)-1)*(cos(x))

Integral of sqrt(2sin(x)-1)*(cos(x)) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
  1                           
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 |  \/ 2*sin(x) - 1 *cos(x) dx
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∫012sin⁡(x)−1cos⁡(x) dx\int\limits_{0}^{1} \sqrt{2 \sin{\left(x \right)} - 1} \cos{\left(x \right)}\, dx 
Integral(sqrt(2*sin(x) - 1)*cos(x), (x, 0, 1))
Detail solution

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

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

    ∫u2 du\int \frac{\sqrt{u}}{2}\, 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:

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

  2. Now simplify:

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

  3. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                                                  
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 |   ______________                 (2*sin(x) - 1)   
 | \/ 2*sin(x) - 1 *cos(x) dx = C + -----------------
 |                                          3        
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∫2sin⁡(x)−1cos⁡(x) dx=C+(2sin⁡(x)−1)323\int \sqrt{2 \sin{\left(x \right)} - 1} \cos{\left(x \right)}\, dx = C + \frac{\left(2 \sin{\left(x \right)} - 1\right)^{\frac{3}{2}}}{3}
Simplify
The graph
1.000.550.600.650.700.750.800.850.900.950.00.5
Plot the graph f(x)
The answer [src] 
    _______________           _______________       
  \/ -1 + 2*sin(1)    I   2*\/ -1 + 2*sin(1) *sin(1)
- ----------------- + - + --------------------------
          3           3               3
−−1+2sin⁡(1)3+2−1+2sin⁡(1)sin⁡(1)3+i3- \frac{\sqrt{-1 + 2 \sin{\left(1 \right)}}}{3} + \frac{2 \sqrt{-1 + 2 \sin{\left(1 \right)}} \sin{\left(1 \right)}}{3} + \frac{i}{3} 
=
= 
    _______________           _______________       
  \/ -1 + 2*sin(1)    I   2*\/ -1 + 2*sin(1) *sin(1)
- ----------------- + - + --------------------------
          3           3               3
−−1+2sin⁡(1)3+2−1+2sin⁡(1)sin⁡(1)3+i3- \frac{\sqrt{-1 + 2 \sin{\left(1 \right)}}}{3} + \frac{2 \sqrt{-1 + 2 \sin{\left(1 \right)}} \sin{\left(1 \right)}}{3} + \frac{i}{3} 
-sqrt(-1 + 2*sin(1))/3 + i/3 + 2*sqrt(-1 + 2*sin(1))*sin(1)/3
Numerical answer [src] 
(0.188144900963713 + 0.333094621053717j)
(0.188144900963713 + 0.333094621053717j)

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

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