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Resolución de integrales/ (x/3)/ sec^3/ tg^2(x/3)sec^3(x/3)dx

Integral de tg^2(x/3)sec^3(x/3)dx dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                   
  /                   
 |                    
 |     2/x\    3/x\   
 |  tan |-|*sec |-| dx
 |      \3/     \3/   
 |                    
/                     
0
$$\int\limits_{0}^{1} \tan^{2}{\left(\frac{x}{3} \right)} \sec^{3}{\left(\frac{x}{3} \right)}\, dx$$ 
Integral(tan(x/3)^2*sec(x/3)^3, (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                              /       /x\\        /        /x\\        /   3/x\      /x\\   
 |                          3*log|1 + sin|-||   3*log|-1 + sin|-||      3*|sin |-| + sin|-||   
 |    2/x\    3/x\               \       \3//        \        \3//        \    \3/      \3//   
 | tan |-|*sec |-| dx = C - ----------------- + ------------------ + --------------------------
 |     \3/     \3/                  16                  16                     2/x\        4/x\
 |                                                                   8 - 16*sin |-| + 8*sin |-|
/                                                                               \3/         \3/
$$\int \tan^{2}{\left(\frac{x}{3} \right)} \sec^{3}{\left(\frac{x}{3} \right)}\, dx = C + \frac{3 \left(\sin^{3}{\left(\frac{x}{3} \right)} + \sin{\left(\frac{x}{3} \right)}\right)}{8 \sin^{4}{\left(\frac{x}{3} \right)} - 16 \sin^{2}{\left(\frac{x}{3} \right)} + 8} + \frac{3 \log{\left(\sin{\left(\frac{x}{3} \right)} - 1 \right)}}{16} - \frac{3 \log{\left(\sin{\left(\frac{x}{3} \right)} + 1 \right)}}{16}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
                                                   /   3                \   
  3*log(1 + sin(1/3))   3*log(1 - sin(1/3))      3*\sin (1/3) + sin(1/3)/   
- ------------------- + ------------------- + ------------------------------
           16                    16                     2             4     
                                              8 - 16*sin (1/3) + 8*sin (1/3)
$$\frac{3 \log{\left(1 - \sin{\left(\frac{1}{3} \right)} \right)}}{16} - \frac{3 \log{\left(\sin{\left(\frac{1}{3} \right)} + 1 \right)}}{16} + \frac{3 \left(\sin^{3}{\left(\frac{1}{3} \right)} + \sin{\left(\frac{1}{3} \right)}\right)}{- 16 \sin^{2}{\left(\frac{1}{3} \right)} + 8 \sin^{4}{\left(\frac{1}{3} \right)} + 8}$$ 
=
= 
                                                   /   3                \   
  3*log(1 + sin(1/3))   3*log(1 - sin(1/3))      3*\sin (1/3) + sin(1/3)/   
- ------------------- + ------------------- + ------------------------------
           16                    16                     2             4     
                                              8 - 16*sin (1/3) + 8*sin (1/3)
$$\frac{3 \log{\left(1 - \sin{\left(\frac{1}{3} \right)} \right)}}{16} - \frac{3 \log{\left(\sin{\left(\frac{1}{3} \right)} + 1 \right)}}{16} + \frac{3 \left(\sin^{3}{\left(\frac{1}{3} \right)} + \sin{\left(\frac{1}{3} \right)}\right)}{- 16 \sin^{2}{\left(\frac{1}{3} \right)} + 8 \sin^{4}{\left(\frac{1}{3} \right)} + 8}$$ 
-3*log(1 + sin(1/3))/16 + 3*log(1 - sin(1/3))/16 + 3*(sin(1/3)^3 + sin(1/3))/(8 - 16*sin(1/3)^2 + 8*sin(1/3)^4)
Respuesta numérica [src] 
0.042975415406588
0.042975415406588

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

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