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1/(cos^2x+2*sinx*cosx)

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  • Similar expressions

  • 1/(cos^2x-2*sinx*cosx)
Solving integrals/ 1/(cos^2x+2*sinx*cosx)

Integral of 1/(cos^2x+2*sinx*cosx) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1                               
  /                               
 |                                
 |                1               
 |  1*------------------------- dx
 |       2                        
 |    cos (x) + 2*sin(x)*cos(x)   
 |                                
/                                 
0
$$\int\limits_{0}^{1} 1 \cdot \frac{1}{2 \sin{\left(x \right)} \cos{\left(x \right)} + \cos^{2}{\left(x \right)}}\, dx$$ 
Integral(1/(cos(x)^2 + 2*sin(x)*cos(x)), (x, 0, 1))
The answer (Indefinite) [src] 
  /                                        /        2/x\        /x\\      /       /x\\      /        /x\\
 |                                      log|-1 + tan |-| - 4*tan|-||   log|1 + tan|-||   log|-1 + tan|-||
 |               1                         \         \2/        \2//      \       \2//      \        \2//
 | 1*------------------------- dx = C + ---------------------------- - --------------- - ----------------
 |      2                                            2                        2                 2        
 |   cos (x) + 2*sin(x)*cos(x)                                                                           
 |                                                                                                       
/
$$\int 1 \cdot \frac{1}{2 \sin{\left(x \right)} \cos{\left(x \right)} + \cos^{2}{\left(x \right)}}\, dx = C - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{2} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{2} + \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} - 4 \tan{\left(\frac{x}{2} \right)} - 1 \right)}}{2}$$
Simplify
The graph
Plot the graph f(x)
The answer [src] 
   /       2                  \                                        
log\1 - tan (1/2) + 4*tan(1/2)/   log(1 - tan(1/2))   log(1 + tan(1/2))
------------------------------- - ----------------- - -----------------
               2                          2                   2
$$- \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{2} - \frac{\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)}}{2} + \frac{\log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 4 \tan{\left(\frac{1}{2} \right)} \right)}}{2}$$ 
=
= 
   /       2                  \                                        
log\1 - tan (1/2) + 4*tan(1/2)/   log(1 - tan(1/2))   log(1 + tan(1/2))
------------------------------- - ----------------- - -----------------
               2                          2                   2
$$- \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{2} - \frac{\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)}}{2} + \frac{\log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 4 \tan{\left(\frac{1}{2} \right)} \right)}}{2}$$
Упростить
Numerical answer [src] 
0.707296992419473
0.707296992419473
The graph
Integral of 1/(cos^2x+2*sinx*cosx) dx

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

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