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Resolución de integrales/ 3cosx/ cosx^2/ cosx^2/(sinx+3cosx)

Integral de cosx^2/(sinx+3cosx) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                     
  /                     
 |                      
 |          2           
 |       cos (x)        
 |  ----------------- dx
 |  sin(x) + 3*cos(x)   
 |                      
/                       
0
∫01cos⁡2(x)sin⁡(x)+3cos⁡(x) dx\int\limits_{0}^{1} \frac{\cos^{2}{\left(x \right)}}{\sin{\left(x \right)} + 3 \cos{\left(x \right)}}\, dx 
Integral(cos(x)^2/(sin(x) + 3*cos(x)), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                                             /        ____         \             /        ____         \                     /        ____         \                     /        ____         \
 |                                                          /x\         ____    |  1   \/ 10       /x\|     ____    |  1   \/ 10       /x\|     ____    2/x\    |  1   \/ 10       /x\|     ____    2/x\    |  1   \/ 10       /x\|
 |         2                                          60*tan|-|       \/ 10 *log|- - + ------ + tan|-||   \/ 10 *log|- - - ------ + tan|-||   \/ 10 *tan |-|*log|- - + ------ + tan|-||   \/ 10 *tan |-|*log|- - - ------ + tan|-||
 |      cos (x)                       20                    \2/                 \  3     3         \2//             \  3     3         \2//              \2/    \  3     3         \2//              \2/    \  3     3         \2//
 | ----------------- dx = C + ----------------- + ----------------- + --------------------------------- - --------------------------------- + ----------------------------------------- - -----------------------------------------
 | sin(x) + 3*cos(x)                       2/x\                2/x\                        2/x\                                2/x\                                    2/x\                                        2/x\            
 |                            100 + 100*tan |-|   100 + 100*tan |-|           100 + 100*tan |-|                   100 + 100*tan |-|                       100 + 100*tan |-|                           100 + 100*tan |-|            
/                                           \2/                 \2/                         \2/                                 \2/                                     \2/                                         \2/
∫cos⁡2(x)sin⁡(x)+3cos⁡(x) dx=C+10log⁡(tan⁡(x2)−13+103)tan⁡2(x2)100tan⁡2(x2)+100+10log⁡(tan⁡(x2)−13+103)100tan⁡2(x2)+100−10log⁡(tan⁡(x2)−103−13)tan⁡2(x2)100tan⁡2(x2)+100−10log⁡(tan⁡(x2)−103−13)100tan⁡2(x2)+100+60tan⁡(x2)100tan⁡2(x2)+100+20100tan⁡2(x2)+100\int \frac{\cos^{2}{\left(x \right)}}{\sin{\left(x \right)} + 3 \cos{\left(x \right)}}\, dx = C + \frac{\sqrt{10} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{1}{3} + \frac{\sqrt{10}}{3} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{100 \tan^{2}{\left(\frac{x}{2} \right)} + 100} + \frac{\sqrt{10} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{1}{3} + \frac{\sqrt{10}}{3} \right)}}{100 \tan^{2}{\left(\frac{x}{2} \right)} + 100} - \frac{\sqrt{10} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{10}}{3} - \frac{1}{3} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{100 \tan^{2}{\left(\frac{x}{2} \right)} + 100} - \frac{\sqrt{10} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{10}}{3} - \frac{1}{3} \right)}}{100 \tan^{2}{\left(\frac{x}{2} \right)} + 100} + \frac{60 \tan{\left(\frac{x}{2} \right)}}{100 \tan^{2}{\left(\frac{x}{2} \right)} + 100} + \frac{20}{100 \tan^{2}{\left(\frac{x}{2} \right)} + 100}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.900.000.50
Trazar el gráfico f(x)
Respuesta [src] 
                                                            /        ____\          /          /      ____\\             /        ____           \          /          /                 ____\\                       /        ____           \                    /          /                 ____\\
                                                    ____    |  1   \/ 10 |     ____ |          |1   \/ 10 ||     ____    |  1   \/ 10            |     ____ |          |1              \/ 10 ||     ____    2         |  1   \/ 10            |     ____    2      |          |1              \/ 10 ||
                                                  \/ 10 *log|- - + ------|   \/ 10 *|pi*I + log|- + ------||   \/ 10 *log|- - + ------ + tan(1/2)|   \/ 10 *|pi*I + log|- - tan(1/2) + ------||   \/ 10 *tan (1/2)*log|- - + ------ + tan(1/2)|   \/ 10 *tan (1/2)*|pi*I + log|- - tan(1/2) + ------||
  1            20               60*tan(1/2)                 \  3     3   /          \          \3     3   //             \  3     3              /          \          \3                3   //                       \  3     3              /                    \          \3                3   //
- - + ------------------- + ------------------- - ------------------------ + ------------------------------- + ----------------------------------- - ------------------------------------------ + --------------------------------------------- - ----------------------------------------------------
  5                2                     2                  100                            100                                      2                                        2                                              2                                                  2                      
      100 + 100*tan (1/2)   100 + 100*tan (1/2)                                                                        100 + 100*tan (1/2)                      100 + 100*tan (1/2)                            100 + 100*tan (1/2)                                100 + 100*tan (1/2)
−15+10log⁡(−13+tan⁡(12)+103)tan⁡2(12)100tan⁡2(12)+100+10log⁡(−13+tan⁡(12)+103)100tan⁡2(12)+100−10log⁡(−13+103)100+20100tan⁡2(12)+100+60tan⁡(12)100tan⁡2(12)+100−10(log⁡(−tan⁡(12)+13+103)+iπ)100tan⁡2(12)+100−10(log⁡(−tan⁡(12)+13+103)+iπ)tan⁡2(12)100tan⁡2(12)+100+10(log⁡(13+103)+iπ)100- \frac{1}{5} + \frac{\sqrt{10} \log{\left(- \frac{1}{3} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{10}}{3} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{100 \tan^{2}{\left(\frac{1}{2} \right)} + 100} + \frac{\sqrt{10} \log{\left(- \frac{1}{3} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{10}}{3} \right)}}{100 \tan^{2}{\left(\frac{1}{2} \right)} + 100} - \frac{\sqrt{10} \log{\left(- \frac{1}{3} + \frac{\sqrt{10}}{3} \right)}}{100} + \frac{20}{100 \tan^{2}{\left(\frac{1}{2} \right)} + 100} + \frac{60 \tan{\left(\frac{1}{2} \right)}}{100 \tan^{2}{\left(\frac{1}{2} \right)} + 100} - \frac{\sqrt{10} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{1}{3} + \frac{\sqrt{10}}{3} \right)} + i \pi\right)}{100 \tan^{2}{\left(\frac{1}{2} \right)} + 100} - \frac{\sqrt{10} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{1}{3} + \frac{\sqrt{10}}{3} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{100 \tan^{2}{\left(\frac{1}{2} \right)} + 100} + \frac{\sqrt{10} \left(\log{\left(\frac{1}{3} + \frac{\sqrt{10}}{3} \right)} + i \pi\right)}{100} 
=
= 
                                                            /        ____\          /          /      ____\\             /        ____           \          /          /                 ____\\                       /        ____           \                    /          /                 ____\\
                                                    ____    |  1   \/ 10 |     ____ |          |1   \/ 10 ||     ____    |  1   \/ 10            |     ____ |          |1              \/ 10 ||     ____    2         |  1   \/ 10            |     ____    2      |          |1              \/ 10 ||
                                                  \/ 10 *log|- - + ------|   \/ 10 *|pi*I + log|- + ------||   \/ 10 *log|- - + ------ + tan(1/2)|   \/ 10 *|pi*I + log|- - tan(1/2) + ------||   \/ 10 *tan (1/2)*log|- - + ------ + tan(1/2)|   \/ 10 *tan (1/2)*|pi*I + log|- - tan(1/2) + ------||
  1            20               60*tan(1/2)                 \  3     3   /          \          \3     3   //             \  3     3              /          \          \3                3   //                       \  3     3              /                    \          \3                3   //
- - + ------------------- + ------------------- - ------------------------ + ------------------------------- + ----------------------------------- - ------------------------------------------ + --------------------------------------------- - ----------------------------------------------------
  5                2                     2                  100                            100                                      2                                        2                                              2                                                  2                      
      100 + 100*tan (1/2)   100 + 100*tan (1/2)                                                                        100 + 100*tan (1/2)                      100 + 100*tan (1/2)                            100 + 100*tan (1/2)                                100 + 100*tan (1/2)
−15+10log⁡(−13+tan⁡(12)+103)tan⁡2(12)100tan⁡2(12)+100+10log⁡(−13+tan⁡(12)+103)100tan⁡2(12)+100−10log⁡(−13+103)100+20100tan⁡2(12)+100+60tan⁡(12)100tan⁡2(12)+100−10(log⁡(−tan⁡(12)+13+103)+iπ)100tan⁡2(12)+100−10(log⁡(−tan⁡(12)+13+103)+iπ)tan⁡2(12)100tan⁡2(12)+100+10(log⁡(13+103)+iπ)100- \frac{1}{5} + \frac{\sqrt{10} \log{\left(- \frac{1}{3} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{10}}{3} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{100 \tan^{2}{\left(\frac{1}{2} \right)} + 100} + \frac{\sqrt{10} \log{\left(- \frac{1}{3} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{10}}{3} \right)}}{100 \tan^{2}{\left(\frac{1}{2} \right)} + 100} - \frac{\sqrt{10} \log{\left(- \frac{1}{3} + \frac{\sqrt{10}}{3} \right)}}{100} + \frac{20}{100 \tan^{2}{\left(\frac{1}{2} \right)} + 100} + \frac{60 \tan{\left(\frac{1}{2} \right)}}{100 \tan^{2}{\left(\frac{1}{2} \right)} + 100} - \frac{\sqrt{10} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{1}{3} + \frac{\sqrt{10}}{3} \right)} + i \pi\right)}{100 \tan^{2}{\left(\frac{1}{2} \right)} + 100} - \frac{\sqrt{10} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{1}{3} + \frac{\sqrt{10}}{3} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{100 \tan^{2}{\left(\frac{1}{2} \right)} + 100} + \frac{\sqrt{10} \left(\log{\left(\frac{1}{3} + \frac{\sqrt{10}}{3} \right)} + i \pi\right)}{100} 
-1/5 + 20/(100 + 100*tan(1/2)^2) + 60*tan(1/2)/(100 + 100*tan(1/2)^2) - sqrt(10)*log(-1/3 + sqrt(10)/3)/100 + sqrt(10)*(pi*i + log(1/3 + sqrt(10)/3))/100 + sqrt(10)*log(-1/3 + sqrt(10)/3 + tan(1/2))/(100 + 100*tan(1/2)^2) - sqrt(10)*(pi*i + log(1/3 - tan(1/2) + sqrt(10)/3))/(100 + 100*tan(1/2)^2) + sqrt(10)*tan(1/2)^2*log(-1/3 + sqrt(10)/3 + tan(1/2))/(100 + 100*tan(1/2)^2) - sqrt(10)*tan(1/2)^2*(pi*i + log(1/3 - tan(1/2) + sqrt(10)/3))/(100 + 100*tan(1/2)^2)
Respuesta numérica [src] 
0.240137697186836
0.240137697186836

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

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