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sin(2*x)/(sin(x)²+2)
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Similar expressions
sin(2*x)/(sin(x)^2-2)
sin(2*x)/(sinx^2+2)

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

Integral of sin(2*x)/(sin(x)^2+2) dx

The solution

You have entered [src]

   0               
   /               
  |                
  |    sin(2*x)    
  |  ----------- dx
  |     2          
  |  sin (x) + 2   
  |                
 /                 
-pi                
----               
 2

$$\int\limits_{- \frac{\pi}{2}}^{0} \frac{\sin{\left(2 x \right)}}{\sin^{2}{\left(x \right)} + 2}\, dx$$

Integral(sin(2*x)/(sin(x)^2 + 2), (x, -pi/2, 0))

Detail solution

There are multiple ways to do this integral.
Method #1
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)} + 2}\, dx = 2 \int \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)} + 2}\, dx$
  1. Let $u = \sin^{2}{\left(x \right)} + 2$ .
    
    Then let $du = 2 \sin{\left(x \right)} \cos{\left(x \right)} dx$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{1}{2 u}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{2}$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $\frac{\log{\left(u \right)}}{2}$
    Now substitute $u$ back in:
    
    $\frac{\log{\left(\sin^{2}{\left(x \right)} + 2 \right)}}{2}$
  So, the result is: $\log{\left(\sin^{2}{\left(x \right)} + 2 \right)}$
Method #2
1. Rewrite the integrand:
  
  $\frac{\sin{\left(2 x \right)}}{\sin^{2}{\left(x \right)} + 2} = \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)} + 2}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)} + 2}\, dx = 2 \int \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)} + 2}\, dx$
  1. Let $u = \sin^{2}{\left(x \right)} + 2$ .
    
    Then let $du = 2 \sin{\left(x \right)} \cos{\left(x \right)} dx$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{1}{2 u}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{2}$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $\frac{\log{\left(u \right)}}{2}$
    Now substitute $u$ back in:
    
    $\frac{\log{\left(\sin^{2}{\left(x \right)} + 2 \right)}}{2}$
  So, the result is: $\log{\left(\sin^{2}{\left(x \right)} + 2 \right)}$
Now simplify:

$\log{\left(\sin^{2}{\left(x \right)} + 2 \right)}$
Add the constant of integration:

$\log{\left(\sin^{2}{\left(x \right)} + 2 \right)}+ \mathrm{constant}$

The answer is:

$\log{\left(\sin^{2}{\left(x \right)} + 2 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                     
 |                                      
 |   sin(2*x)              /   2       \
 | ----------- dx = C + log\sin (x) + 2/
 |    2                                 
 | sin (x) + 2                          
 |                                      
/

$$\int \frac{\sin{\left(2 x \right)}}{\sin^{2}{\left(x \right)} + 2}\, dx = C + \log{\left(\sin^{2}{\left(x \right)} + 2 \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-log(3) + log(2)

$$- \log{\left(3 \right)} + \log{\left(2 \right)}$$

-log(3) + log(2)

$$- \log{\left(3 \right)} + \log{\left(2 \right)}$$

-log(3) + log(2)

Numerical answer [src]

-0.405465108108164

-0.405465108108164

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

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Identical expressions

Similar expressions

Integral of sin(2*x)/(sin(x)^2+2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2