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cos(x)/(five - two sin(x)-sin^2(x))
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Solving integrals/ cos(x)/(5-2sin(x)-sin^2(x))

Integral of cos(x)/(5-2sin(x)-sin^2(x)) dx

The solution

You have entered [src]

 pi                          
 --                          
 2                           
  /                          
 |                           
 |          cos(x)           
 |  ---------------------- dx
 |                    2      
 |  5 - 2*sin(x) - sin (x)   
 |                           
/                            
0

$$\int\limits_{0}^{\frac{\pi}{2}} \frac{\cos{\left(x \right)}}{\left(5 - 2 \sin{\left(x \right)}\right) - \sin^{2}{\left(x \right)}}\, dx$$

Integral(cos(x)/(5 - 2*sin(x) - sin(x)^2), (x, 0, pi/2))

Detail solution

Rewrite the integrand:

$\frac{\cos{\left(x \right)}}{\left(5 - 2 \sin{\left(x \right)}\right) - \sin^{2}{\left(x \right)}} = - \frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)} + 2 \sin{\left(x \right)} - 5}$
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)} + 2 \sin{\left(x \right)} - 5}\right)\, dx = - \int \frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)} + 2 \sin{\left(x \right)} - 5}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $- \frac{\sqrt{6} \log{\left(\sin{\left(x \right)} + 1 + \sqrt{6} \right)}}{12} + \frac{\sqrt{6} \log{\left(\sin{\left(x \right)} - \sqrt{6} + 1 \right)}}{12}$
So, the result is: $\frac{\sqrt{6} \log{\left(\sin{\left(x \right)} + 1 + \sqrt{6} \right)}}{12} - \frac{\sqrt{6} \log{\left(\sin{\left(x \right)} - \sqrt{6} + 1 \right)}}{12}$
Now simplify:

$\frac{\sqrt{6} \left(\log{\left(\sin{\left(x \right)} + 1 + \sqrt{6} \right)} - \log{\left(\sin{\left(x \right)} - \sqrt{6} + 1 \right)}\right)}{12}$
Add the constant of integration:

$\frac{\sqrt{6} \left(\log{\left(\sin{\left(x \right)} + 1 + \sqrt{6} \right)} - \log{\left(\sin{\left(x \right)} - \sqrt{6} + 1 \right)}\right)}{12}+ \mathrm{constant}$

The answer is:

$\frac{\sqrt{6} \left(\log{\left(\sin{\left(x \right)} + 1 + \sqrt{6} \right)} - \log{\left(\sin{\left(x \right)} - \sqrt{6} + 1 \right)}\right)}{12}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                             
 |                                   ___    /      ___         \     ___    /      ___         \
 |         cos(x)                  \/ 6 *log\1 - \/ 6  + sin(x)/   \/ 6 *log\1 + \/ 6  + sin(x)/
 | ---------------------- dx = C - ----------------------------- + -----------------------------
 |                   2                           12                              12             
 | 5 - 2*sin(x) - sin (x)                                                                       
 |                                                                                              
/

$$\int \frac{\cos{\left(x \right)}}{\left(5 - 2 \sin{\left(x \right)}\right) - \sin^{2}{\left(x \right)}}\, dx = C + \frac{\sqrt{6} \log{\left(\sin{\left(x \right)} + 1 + \sqrt{6} \right)}}{12} - \frac{\sqrt{6} \log{\left(\sin{\left(x \right)} - \sqrt{6} + 1 \right)}}{12}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

    ___ /          /       ___\\     ___    /      ___\     ___ /          /       ___\\     ___    /      ___\
  \/ 6 *\pi*I + log\-2 + \/ 6 //   \/ 6 *log\1 + \/ 6 /   \/ 6 *\pi*I + log\-1 + \/ 6 //   \/ 6 *log\2 + \/ 6 /
- ------------------------------ - -------------------- + ------------------------------ + --------------------
                12                          12                          12                          12

$$- \frac{\sqrt{6} \log{\left(1 + \sqrt{6} \right)}}{12} + \frac{\sqrt{6} \log{\left(2 + \sqrt{6} \right)}}{12} - \frac{\sqrt{6} \left(\log{\left(-2 + \sqrt{6} \right)} + i \pi\right)}{12} + \frac{\sqrt{6} \left(\log{\left(-1 + \sqrt{6} \right)} + i \pi\right)}{12}$$

    ___ /          /       ___\\     ___    /      ___\     ___ /          /       ___\\     ___    /      ___\
  \/ 6 *\pi*I + log\-2 + \/ 6 //   \/ 6 *log\1 + \/ 6 /   \/ 6 *\pi*I + log\-1 + \/ 6 //   \/ 6 *log\2 + \/ 6 /
- ------------------------------ - -------------------- + ------------------------------ + --------------------
                12                          12                          12                          12

-sqrt(6)*(pi*i + log(-2 + sqrt(6)))/12 - sqrt(6)*log(1 + sqrt(6))/12 + sqrt(6)*(pi*i + log(-1 + sqrt(6)))/12 + sqrt(6)*log(2 + sqrt(6))/12

Numerical answer [src]

0.290962015103402

0.290962015103402

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

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Identical expressions

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Integral of cos(x)/(5-2sin(x)-sin^2(x)) dx

Limits of integration:

The graph:

Piecewise:

The solution