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

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

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

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

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

$\int \frac{\sqrt{u}}{2}\, du$
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:

$\frac{\left(2 \sin{\left(x \right)} - 1\right)^{\frac{3}{2}}}{3}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

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.

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Identical expressions

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

Limits of integration:

The graph:

Piecewise:

The solution