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cos2x/(1+sin2x)
Solving integrals/ cos2x/(1+sin2x)

Integral of cos2x/(1+sin2x) dx

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The solution

You have entered [src] 
  1                
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 |    cos(2*x)     
 |  ------------ dx
 |  1 + sin(2*x)   
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∫01cos⁡(2x)sin⁡(2x)+1 dx\int\limits_{0}^{1} \frac{\cos{\left(2 x \right)}}{\sin{\left(2 x \right)} + 1}\, dx 
Integral(cos(2*x)/(1 + sin(2*x)), (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=2xu = 2 x.

      Then let du=2dxdu = 2 dx and substitute dudu:

      ∫cos⁡(u)2sin⁡(u)+2 du\int \frac{\cos{\left(u \right)}}{2 \sin{\left(u \right)} + 2}\, du

      1. Let u=2sin⁡(u)+2u = 2 \sin{\left(u \right)} + 2.

        Then let du=2cos⁡(u)dudu = 2 \cos{\left(u \right)} du and substitute du2\frac{du}{2}:

        ∫14u du\int \frac{1}{4 u}\, du

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

          ∫12u du=∫1u du2\int \frac{1}{2 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⁡(2sin⁡(u)+2)2\frac{\log{\left(2 \sin{\left(u \right)} + 2 \right)}}{2}

      Now substitute uu back in:

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

    Method #2

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

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

      ∫14u du\int \frac{1}{4 u}\, du

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

        ∫12u du=∫1u du2\int \frac{1}{2 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⁡(2x)+1)2\frac{\log{\left(\sin{\left(2 x \right)} + 1 \right)}}{2}

  2. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                                         
 |                                          
 |   cos(2*x)            log(2 + 2*sin(2*x))
 | ------------ dx = C + -------------------
 | 1 + sin(2*x)                   2         
 |                                          
/
∫cos⁡(2x)sin⁡(2x)+1 dx=C+log⁡(2sin⁡(2x)+2)2\int \frac{\cos{\left(2 x \right)}}{\sin{\left(2 x \right)} + 1}\, dx = C + \frac{\log{\left(2 \sin{\left(2 x \right)} + 2 \right)}}{2}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.902-1
Plot the graph f(x)
The answer [src] 
log(1 + sin(2))
---------------
       2
log⁡(sin⁡(2)+1)2\frac{\log{\left(\sin{\left(2 \right)} + 1 \right)}}{2} 
=
= 
log(1 + sin(2))
---------------
       2
log⁡(sin⁡(2)+1)2\frac{\log{\left(\sin{\left(2 \right)} + 1 \right)}}{2}
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Numerical answer [src] 
0.323367667515383
0.323367667515383
The graph
Integral of cos2x/(1+sin2x) dx

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

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