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sin(x)/(cos(x)+2)
Solving integrals/ sin(x)/(cos(x)+2)

Integral of sin(x)/(cos(x)+2) dx

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The solution

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

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

    Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

    ∫(−1u) du\int \left(- \frac{1}{u}\right)\, du

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

      ∫1u du=−∫1u du\int \frac{1}{u}\, du = - \int \frac{1}{u}\, du

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

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

    Now substitute uu back in:

    −log⁡(cos⁡(x)+2)- \log{\left(\cos{\left(x \right)} + 2 \right)}

  2. Now simplify:

    −log⁡(cos⁡(x)+2)- \log{\left(\cos{\left(x \right)} + 2 \right)}

  3. Add the constant of integration:

    −log⁡(cos⁡(x)+2)+constant- \log{\left(\cos{\left(x \right)} + 2 \right)}+ \mathrm{constant}

The answer is:

−log⁡(cos⁡(x)+2)+constant- \log{\left(\cos{\left(x \right)} + 2 \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                   
 |                                    
 |   sin(x)                           
 | ---------- dx = C - log(cos(x) + 2)
 | cos(x) + 2                         
 |                                    
/
∫sin⁡(x)cos⁡(x)+2 dx=C−log⁡(cos⁡(x)+2)\int \frac{\sin{\left(x \right)}}{\cos{\left(x \right)} + 2}\, dx = C - \log{\left(\cos{\left(x \right)} + 2 \right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.902-2
Plot the graph f(x)
The answer [src] 
-log(2 + cos(1)) + log(3)
−log⁡(cos⁡(1)+2)+log⁡(3)- \log{\left(\cos{\left(1 \right)} + 2 \right)} + \log{\left(3 \right)} 
=
= 
-log(2 + cos(1)) + log(3)
−log⁡(cos⁡(1)+2)+log⁡(3)- \log{\left(\cos{\left(1 \right)} + 2 \right)} + \log{\left(3 \right)} 
-log(2 + cos(1)) + log(3)
Numerical answer [src] 
0.166329196661429
0.166329196661429
The graph
Integral of sin(x)/(cos(x)+2) dx

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

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