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1/(sin^2x+2cos^2x)

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  • 1/(sin^2x-2cos^2x)
Solving integrals/ 1/(sin^2x+2cos^2x)

Integral of 1/(sin^2x+2cos^2x) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
  1                       
  /                       
 |                        
 |           1            
 |  ------------------- dx
 |     2           2      
 |  sin (x) + 2*cos (x)   
 |                        
/                         
0
$$\int\limits_{0}^{1} \frac{1}{\sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}}\, dx$$ 
Integral(1/(sin(x)^2 + 2*cos(x)^2), (x, 0, 1))
The answer (Indefinite) [src] 
                                      /        /x   pi\                         \         /        /x   pi\                          \
                                      |        |- - --|                         |         |        |- - --|                          |
  /                               ___ |        |2   2 |       /      ___    /x\\|     ___ |        |2   2 |       /       ___    /x\\|
 |                              \/ 2 *|pi*floor|------| + atan|1 + \/ 2 *tan|-|||   \/ 2 *|pi*floor|------| + atan|-1 + \/ 2 *tan|-|||
 |          1                         \        \  pi  /       \             \2///         \        \  pi  /       \              \2///
 | ------------------- dx = C + ------------------------------------------------- + --------------------------------------------------
 |    2           2                                     2                                                   2                         
 | sin (x) + 2*cos (x)                                                                                                                
 |                                                                                                                                    
/
$$\int \frac{1}{\sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}}\, dx = C + \frac{\sqrt{2} \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{2} + \frac{\sqrt{2} \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{2}$$
Simplify
The graph
Plot the graph f(x)
The answer [src] 
             ___ /          /      ___         \\     ___ /          /      ___         \\
     ___   \/ 2 *\-pi - atan\1 - \/ 2 *tan(1/2)//   \/ 2 *\-pi + atan\1 + \/ 2 *tan(1/2)//
pi*\/ 2  + -------------------------------------- + --------------------------------------
                             2                                        2
$$\frac{\sqrt{2} \left(- \pi - \operatorname{atan}{\left(- \sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right)}{2} + \frac{\sqrt{2} \left(- \pi + \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right)}{2} + \sqrt{2} \pi$$ 
=
= 
             ___ /          /      ___         \\     ___ /          /      ___         \\
     ___   \/ 2 *\-pi - atan\1 - \/ 2 *tan(1/2)//   \/ 2 *\-pi + atan\1 + \/ 2 *tan(1/2)//
pi*\/ 2  + -------------------------------------- + --------------------------------------
                             2                                        2
$$\frac{\sqrt{2} \left(- \pi - \operatorname{atan}{\left(- \sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right)}{2} + \frac{\sqrt{2} \left(- \pi + \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right)}{2} + \sqrt{2} \pi$$ 
pi*sqrt(2) + sqrt(2)*(-pi - atan(1 - sqrt(2)*tan(1/2)))/2 + sqrt(2)*(-pi + atan(1 + sqrt(2)*tan(1/2)))/2
Numerical answer [src] 
0.589407539463719
0.589407539463719
The graph
Integral of 1/(sin^2x+2cos^2x) dx

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

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