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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

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

    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. Now simplify:

  3. Add the constant of integration:


The answer is:

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