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  • sin(2*x)/(sin(x)^2+2)dx

  • Similar expressions

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

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   0               
   /               
  |                
  |    sin(2*x)    
  |  ----------- dx
  |     2          
  |  sin (x) + 2   
  |                
 /                 
-pi                
----               
 2
∫−π20sin⁡(2x)sin⁡2(x)+2 dx\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

  1. 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:

      ∫2sin⁡(x)cos⁡(x)sin⁡2(x)+2 dx=2∫sin⁡(x)cos⁡(x)sin⁡2(x)+2 dx\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(x)+2u = \sin^{2}{\left(x \right)} + 2.

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

        ∫12u du\int \frac{1}{2 u}\, du

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

          ∫1u du=∫1u du2\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{2}

          1. The integral of 1u\frac{1}{u} is log⁡(u)\log{\left(u \right)}.

          So, the result is: log⁡(u)2\frac{\log{\left(u \right)}}{2}

        Now substitute uu back in:

        log⁡(sin⁡2(x)+2)2\frac{\log{\left(\sin^{2}{\left(x \right)} + 2 \right)}}{2}

      So, the result is: log⁡(sin⁡2(x)+2)\log{\left(\sin^{2}{\left(x \right)} + 2 \right)}

    Method #2

    1. Rewrite the integrand:

      sin⁡(2x)sin⁡2(x)+2=2sin⁡(x)cos⁡(x)sin⁡2(x)+2\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:

      ∫2sin⁡(x)cos⁡(x)sin⁡2(x)+2 dx=2∫sin⁡(x)cos⁡(x)sin⁡2(x)+2 dx\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(x)+2u = \sin^{2}{\left(x \right)} + 2.

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

        ∫12u du\int \frac{1}{2 u}\, du

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

          ∫1u du=∫1u du2\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{2}

          1. The integral of 1u\frac{1}{u} is log⁡(u)\log{\left(u \right)}.

          So, the result is: log⁡(u)2\frac{\log{\left(u \right)}}{2}

        Now substitute uu back in:

        log⁡(sin⁡2(x)+2)2\frac{\log{\left(\sin^{2}{\left(x \right)} + 2 \right)}}{2}

      So, the result is: log⁡(sin⁡2(x)+2)\log{\left(\sin^{2}{\left(x \right)} + 2 \right)}

  2. Now simplify:

    log⁡(sin⁡2(x)+2)\log{\left(\sin^{2}{\left(x \right)} + 2 \right)}

  3. Add the constant of integration:

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

The answer is:

log⁡(sin⁡2(x)+2)+constant\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                          
 |                                      
/
∫sin⁡(2x)sin⁡2(x)+2 dx=C+log⁡(sin⁡2(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
-1.5-1.4-1.3-1.2-1.1-1.0-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.10.02-2
Plot the graph f(x)
The answer [src] 
-log(3) + log(2)
−log⁡(3)+log⁡(2)- \log{\left(3 \right)} + \log{\left(2 \right)} 
=
= 
-log(3) + log(2)
−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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