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Resolución de integrales/ cos^2/ x/cos^2(3x)

Integral de x/cos^2(3x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 pi             
 --             
 9              
  /             
 |              
 |      x       
 |  --------- dx
 |     2        
 |  cos (3*x)   
 |              
/               
0
∫0π9xcos⁡2(3x) dx\int\limits_{0}^{\frac{\pi}{9}} \frac{x}{\cos^{2}{\left(3 x \right)}}\, dx 
Integral(x/cos(3*x)^2, (x, 0, pi/9))
Respuesta (Indefinida) [src] 
  /                      /       2/3*x\\      /       /3*x\\      /        /3*x\\      2/3*x\    /       /3*x\\      2/3*x\    /        /3*x\\      2/3*x\    /       2/3*x\\            /3*x\  
 |                    log|1 + tan |---||   log|1 + tan|---||   log|-1 + tan|---||   tan |---|*log|1 + tan|---||   tan |---|*log|-1 + tan|---||   tan |---|*log|1 + tan |---||     6*x*tan|---|  
 |     x                 \        \ 2 //      \       \ 2 //      \        \ 2 //       \ 2 /    \       \ 2 //       \ 2 /    \        \ 2 //       \ 2 /    \        \ 2 //            \ 2 /  
 | --------- dx = C + ------------------ - ----------------- - ------------------ + --------------------------- + ---------------------------- - ---------------------------- - ----------------
 |    2                          2/3*x\               2/3*x\              2/3*x\                    2/3*x\                        2/3*x\                         2/3*x\                   2/3*x\
 | cos (3*x)           -9 + 9*tan |---|     -9 + 9*tan |---|    -9 + 9*tan |---|          -9 + 9*tan |---|              -9 + 9*tan |---|               -9 + 9*tan |---|         -9 + 9*tan |---|
 |                                \ 2 /                \ 2 /               \ 2 /                     \ 2 /                         \ 2 /                          \ 2 /                    \ 2 /
/
∫xcos⁡2(3x) dx=C−6xtan⁡(3x2)9tan⁡2(3x2)−9+log⁡(tan⁡(3x2)−1)tan⁡2(3x2)9tan⁡2(3x2)−9−log⁡(tan⁡(3x2)−1)9tan⁡2(3x2)−9+log⁡(tan⁡(3x2)+1)tan⁡2(3x2)9tan⁡2(3x2)−9−log⁡(tan⁡(3x2)+1)9tan⁡2(3x2)−9−log⁡(tan⁡2(3x2)+1)tan⁡2(3x2)9tan⁡2(3x2)−9+log⁡(tan⁡2(3x2)+1)9tan⁡2(3x2)−9\int \frac{x}{\cos^{2}{\left(3 x \right)}}\, dx = C - \frac{6 x \tan{\left(\frac{3 x}{2} \right)}}{9 \tan^{2}{\left(\frac{3 x}{2} \right)} - 9} + \frac{\log{\left(\tan{\left(\frac{3 x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{3 x}{2} \right)}}{9 \tan^{2}{\left(\frac{3 x}{2} \right)} - 9} - \frac{\log{\left(\tan{\left(\frac{3 x}{2} \right)} - 1 \right)}}{9 \tan^{2}{\left(\frac{3 x}{2} \right)} - 9} + \frac{\log{\left(\tan{\left(\frac{3 x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{3 x}{2} \right)}}{9 \tan^{2}{\left(\frac{3 x}{2} \right)} - 9} - \frac{\log{\left(\tan{\left(\frac{3 x}{2} \right)} + 1 \right)}}{9 \tan^{2}{\left(\frac{3 x}{2} \right)} - 9} - \frac{\log{\left(\tan^{2}{\left(\frac{3 x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{3 x}{2} \right)}}{9 \tan^{2}{\left(\frac{3 x}{2} \right)} - 9} + \frac{\log{\left(\tan^{2}{\left(\frac{3 x}{2} \right)} + 1 \right)}}{9 \tan^{2}{\left(\frac{3 x}{2} \right)} - 9}
Simplificar
Gráfica
0.000.050.100.150.200.250.3002
Trazar el gráfico f(x)
Respuesta [src] 
                /      ___\      /      ___\           
                |    \/ 3 |      |    \/ 3 |           
             log|1 - -----|   log|1 + -----|        ___
  log(4/3)      \      3  /      \      3  /   pi*\/ 3 
- -------- + -------------- + -------------- + --------
     9             9                9             27
log⁡(1−33)9−log⁡(43)9+log⁡(33+1)9+3π27\frac{\log{\left(1 - \frac{\sqrt{3}}{3} \right)}}{9} - \frac{\log{\left(\frac{4}{3} \right)}}{9} + \frac{\log{\left(\frac{\sqrt{3}}{3} + 1 \right)}}{9} + \frac{\sqrt{3} \pi}{27} 
=
= 
                /      ___\      /      ___\           
                |    \/ 3 |      |    \/ 3 |           
             log|1 - -----|   log|1 + -----|        ___
  log(4/3)      \      3  /      \      3  /   pi*\/ 3 
- -------- + -------------- + -------------- + --------
     9             9                9             27
log⁡(1−33)9−log⁡(43)9+log⁡(33+1)9+3π27\frac{\log{\left(1 - \frac{\sqrt{3}}{3} \right)}}{9} - \frac{\log{\left(\frac{4}{3} \right)}}{9} + \frac{\log{\left(\frac{\sqrt{3}}{3} + 1 \right)}}{9} + \frac{\sqrt{3} \pi}{27} 
-log(4/3)/9 + log(1 - sqrt(3)/3)/9 + log(1 + sqrt(3)/3)/9 + pi*sqrt(3)/27
Respuesta numérica [src] 
0.124516909297141
0.124516909297141

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

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