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Identical expressions
(2tgx+ three)/(sin^2x+2cos^2x)
(2tgx plus 3) divide by ( sinus of squared x plus 2 co sinus of e of squared x)
(2tgx plus three) divide by ( sinus of squared x plus 2 co sinus of e of squared x)
(2tgx+3)/(sin2x+2cos2x)
2tgx+3/sin2x+2cos2x
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(2tgx+3)/(sin to the power of 2x+2cos to the power of 2x)
2tgx+3/sin^2x+2cos^2x
(2tgx+3) divide by (sin^2x+2cos^2x)
(2tgx+3)/(sin^2x+2cos^2x)dx
Similar expressions
(2tgx-3)/(sin^2x+2cos^2x)
(2tgx+3)/(sin^2x-2cos^2x)

Solving integrals/ (2tgx+3)/(sin^2x+2cos^2x)

Integral of (2tgx+3)/(sin^2x+2cos^2x) dx

The solution

You have entered [src]

  1                       
  /                       
 |                        
 |      2*tan(x) + 3      
 |  ------------------- dx
 |     2           2      
 |  sin (x) + 2*cos (x)   
 |                        
/                         
0

$$\int\limits_{0}^{1} \frac{2 \tan{\left(x \right)} + 3}{\sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}}\, dx$$

Integral((2*tan(x) + 3)/(sin(x)^2 + 2*cos(x)^2), (x, 0, 1))

Detail solution

Rewrite the integrand:

$\frac{2 \tan{\left(x \right)} + 3}{\sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}} = \frac{2 \tan{\left(x \right)}}{\sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}} + \frac{3}{\sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{2 \tan{\left(x \right)}}{\sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}}\, dx = 2 \int \frac{\tan{\left(x \right)}}{\sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\int \frac{\tan{\left(x \right)}}{\sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}}\, dx$
  So, the result is: $2 \int \frac{\tan{\left(x \right)}}{\sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}}\, dx$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{3}{\sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}}\, dx = 3 \int \frac{1}{\sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\frac{\sqrt{2} \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{2} + \frac{\sqrt{2} \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{2}$
  So, the result is: $\frac{3 \sqrt{2} \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{2} + \frac{3 \sqrt{2} \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{2}$
The result is: $\frac{3 \sqrt{2} \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{2} + \frac{3 \sqrt{2} \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{2} + 2 \int \frac{\tan{\left(x \right)}}{\sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}}\, dx$
Now simplify:

$\frac{3 \sqrt{2} \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)}}{2} + \frac{3 \sqrt{2} \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{2} + 3 \sqrt{2} \pi \left\lfloor{\frac{x - \pi}{2 \pi}}\right\rfloor - 2 \int \frac{\tan{\left(x \right)}}{\sin^{2}{\left(x \right)} - 2}\, dx$
Add the constant of integration:

$\frac{3 \sqrt{2} \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)}}{2} + \frac{3 \sqrt{2} \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{2} + 3 \sqrt{2} \pi \left\lfloor{\frac{x - \pi}{2 \pi}}\right\rfloor - 2 \int \frac{\tan{\left(x \right)}}{\sin^{2}{\left(x \right)} - 2}\, dx+ \mathrm{constant}$

The answer is:

$\frac{3 \sqrt{2} \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)}}{2} + \frac{3 \sqrt{2} \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{2} + 3 \sqrt{2} \pi \left\lfloor{\frac{x - \pi}{2 \pi}}\right\rfloor - 2 \int \frac{\tan{\left(x \right)}}{\sin^{2}{\left(x \right)} - 2}\, dx+ \mathrm{constant}$

The answer (Indefinite) [src]

                                                                      /        /x   pi\                         \           /        /x   pi\                          \
                                                                      |        |- - --|                         |           |        |- - --|                          |
  /                                 /                             ___ |        |2   2 |       /      ___    /x\\|       ___ |        |2   2 |       /       ___    /x\\|
 |                                 |                          3*\/ 2 *|pi*floor|------| + atan|1 + \/ 2 *tan|-|||   3*\/ 2 *|pi*floor|------| + atan|-1 + \/ 2 *tan|-|||
 |     2*tan(x) + 3                |        tan(x)                    \        \  pi  /       \             \2///           \        \  pi  /       \              \2///
 | ------------------- dx = C + 2* | ------------------- dx + --------------------------------------------------- + ----------------------------------------------------
 |    2           2                |    2           2                                  2                                                     2                          
 | sin (x) + 2*cos (x)             | sin (x) + 2*cos (x)                                                                                                                
 |                                 |                                                                                                                                    
/                                 /

$$\log \left(\tan ^2x+2\right)+{{3\,\arctan \left({{\tan x}\over{ \sqrt{2}}}\right)}\over{\sqrt{2}}}$$

Simplify

The answer [src]

  1                       
  /                       
 |                        
 |      3 + 2*tan(x)      
 |  ------------------- dx
 |     2           2      
 |  sin (x) + 2*cos (x)   
 |                        
/                         
0

$$\log \left(\tan ^21+2\right)+{{3\,\arctan \left({{\sin 1}\over{ \sqrt{2}\,\cos 1}}\right)}\over{\sqrt{2}}}-\log 2$$

  1                       
  /                       
 |                        
 |      3 + 2*tan(x)      
 |  ------------------- dx
 |     2           2      
 |  sin (x) + 2*cos (x)   
 |                        
/                         
0

$$\int\limits_{0}^{1} \frac{2 \tan{\left(x \right)} + 3}{\sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}}\, dx$$

Simplify

Numerical answer [src]

2.56246295705989

2.56246295705989

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

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Identical expressions

Similar expressions

Integral of (2tgx+3)/(sin^2x+2cos^2x) dx

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The solution