Integral de (cosh(x))^3÷(1+sinh(x)) dx
Solución
Respuesta (Indefinida)
[src]
/
| / /x\\ /x\ 2/x\ 3/x\ / 2/x\ /x\\ 2/x\ 2/x\ / 2/x\ /x\\ 4/x\ / /x\\ 4/x\ 4/x\ / 2/x\ /x\\ 2/x\ / /x\\
| 3 4*log|1 + tanh|-|| 2*tanh|-| 2*tanh |-| 2*tanh |-| 2*log|-1 + tanh |-| - 2*tanh|-|| 4*x*tanh |-| 4*tanh |-|*log|-1 + tanh |-| - 2*tanh|-|| 4*tanh |-|*log|1 + tanh|-|| 2*x*tanh |-| 2*tanh |-|*log|-1 + tanh |-| - 2*tanh|-|| 8*tanh |-|*log|1 + tanh|-||
| cosh (x) \ \2// \2/ 2*x \2/ \2/ \ \2/ \2// \2/ \2/ \ \2/ \2// \2/ \ \2// \2/ \2/ \ \2/ \2// \2/ \ \2//
| ----------- dx = C - ------------------------- - ------------------------- + ------------------------- + ------------------------- + ------------------------- + -------------------------------- - ------------------------- - ----------------------------------------- - --------------------------- + ------------------------- + ----------------------------------------- + ---------------------------
| 1 + sinh(x) 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\
| 1 + tanh |-| - 2*tanh |-| 1 + tanh |-| - 2*tanh |-| 1 + tanh |-| - 2*tanh |-| 1 + tanh |-| - 2*tanh |-| 1 + tanh |-| - 2*tanh |-| 1 + tanh |-| - 2*tanh |-| 1 + tanh |-| - 2*tanh |-| 1 + tanh |-| - 2*tanh |-| 1 + tanh |-| - 2*tanh |-| 1 + tanh |-| - 2*tanh |-| 1 + tanh |-| - 2*tanh |-| 1 + tanh |-| - 2*tanh |-|
/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/
$$\int \frac{\cosh^{3}{\left(x \right)}}{\sinh{\left(x \right)} + 1}\, dx = C + \frac{2 x \tanh^{4}{\left(\frac{x}{2} \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} - 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{4 x \tanh^{2}{\left(\frac{x}{2} \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} - 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 x}{\tanh^{4}{\left(\frac{x}{2} \right)} - 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{4 \log{\left(\tanh{\left(\frac{x}{2} \right)} + 1 \right)} \tanh^{4}{\left(\frac{x}{2} \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} - 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{8 \log{\left(\tanh{\left(\frac{x}{2} \right)} + 1 \right)} \tanh^{2}{\left(\frac{x}{2} \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} - 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{4 \log{\left(\tanh{\left(\frac{x}{2} \right)} + 1 \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} - 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \log{\left(\tanh^{2}{\left(\frac{x}{2} \right)} - 2 \tanh{\left(\frac{x}{2} \right)} - 1 \right)} \tanh^{4}{\left(\frac{x}{2} \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} - 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{4 \log{\left(\tanh^{2}{\left(\frac{x}{2} \right)} - 2 \tanh{\left(\frac{x}{2} \right)} - 1 \right)} \tanh^{2}{\left(\frac{x}{2} \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} - 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \log{\left(\tanh^{2}{\left(\frac{x}{2} \right)} - 2 \tanh{\left(\frac{x}{2} \right)} - 1 \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} - 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \tanh^{3}{\left(\frac{x}{2} \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} - 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \tanh^{2}{\left(\frac{x}{2} \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} - 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{2 \tanh{\left(\frac{x}{2} \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} - 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1}$$
2 3 4 / / 2 \\ 2 / / 2 \\ 4 4 / / 2 \\ 2
2 4*log(1 + tanh(1/2)) 2*tanh (1/2) 2*tanh(1/2) 2*tanh (1/2) 2*tanh (1/2) 2*\pi*I + log\1 - tanh (1/2) + 2*tanh(1/2)// 4*tanh (1/2)*\pi*I + log\1 - tanh (1/2) + 2*tanh(1/2)// 4*tanh (1/2)*log(1 + tanh(1/2)) 2*tanh (1/2)*\pi*I + log\1 - tanh (1/2) + 2*tanh(1/2)// 8*tanh (1/2)*log(1 + tanh(1/2))
----------------------------- - ----------------------------- - 2*pi*I - ----------------------------- - ----------------------------- + ----------------------------- + ----------------------------- + -------------------------------------------- - ------------------------------------------------------- - ------------------------------- + ------------------------------------------------------- + -------------------------------
4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2
1 + tanh (1/2) - 2*tanh (1/2) 1 + tanh (1/2) - 2*tanh (1/2) 1 + tanh (1/2) - 2*tanh (1/2) 1 + tanh (1/2) - 2*tanh (1/2) 1 + tanh (1/2) - 2*tanh (1/2) 1 + tanh (1/2) - 2*tanh (1/2) 1 + tanh (1/2) - 2*tanh (1/2) 1 + tanh (1/2) - 2*tanh (1/2) 1 + tanh (1/2) - 2*tanh (1/2) 1 + tanh (1/2) - 2*tanh (1/2) 1 + tanh (1/2) - 2*tanh (1/2)
$$- \frac{4 \log{\left(\tanh{\left(\frac{1}{2} \right)} + 1 \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} - \frac{2 \tanh{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} - \frac{2 \tanh^{2}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} - \frac{4 \log{\left(\tanh{\left(\frac{1}{2} \right)} + 1 \right)} \tanh^{4}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \tanh^{4}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \tanh^{3}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} + \frac{8 \log{\left(\tanh{\left(\frac{1}{2} \right)} + 1 \right)} \tanh^{2}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} + \frac{2}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} - 2 i \pi - \frac{4 \left(\log{\left(- \tanh^{2}{\left(\frac{1}{2} \right)} + 2 \tanh{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right) \tanh^{2}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \left(\log{\left(- \tanh^{2}{\left(\frac{1}{2} \right)} + 2 \tanh{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right) \tanh^{4}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \left(\log{\left(- \tanh^{2}{\left(\frac{1}{2} \right)} + 2 \tanh{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right)}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1}$$
=
2 3 4 / / 2 \\ 2 / / 2 \\ 4 4 / / 2 \\ 2
2 4*log(1 + tanh(1/2)) 2*tanh (1/2) 2*tanh(1/2) 2*tanh (1/2) 2*tanh (1/2) 2*\pi*I + log\1 - tanh (1/2) + 2*tanh(1/2)// 4*tanh (1/2)*\pi*I + log\1 - tanh (1/2) + 2*tanh(1/2)// 4*tanh (1/2)*log(1 + tanh(1/2)) 2*tanh (1/2)*\pi*I + log\1 - tanh (1/2) + 2*tanh(1/2)// 8*tanh (1/2)*log(1 + tanh(1/2))
----------------------------- - ----------------------------- - 2*pi*I - ----------------------------- - ----------------------------- + ----------------------------- + ----------------------------- + -------------------------------------------- - ------------------------------------------------------- - ------------------------------- + ------------------------------------------------------- + -------------------------------
4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2
1 + tanh (1/2) - 2*tanh (1/2) 1 + tanh (1/2) - 2*tanh (1/2) 1 + tanh (1/2) - 2*tanh (1/2) 1 + tanh (1/2) - 2*tanh (1/2) 1 + tanh (1/2) - 2*tanh (1/2) 1 + tanh (1/2) - 2*tanh (1/2) 1 + tanh (1/2) - 2*tanh (1/2) 1 + tanh (1/2) - 2*tanh (1/2) 1 + tanh (1/2) - 2*tanh (1/2) 1 + tanh (1/2) - 2*tanh (1/2) 1 + tanh (1/2) - 2*tanh (1/2)
$$- \frac{4 \log{\left(\tanh{\left(\frac{1}{2} \right)} + 1 \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} - \frac{2 \tanh{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} - \frac{2 \tanh^{2}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} - \frac{4 \log{\left(\tanh{\left(\frac{1}{2} \right)} + 1 \right)} \tanh^{4}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \tanh^{4}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \tanh^{3}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} + \frac{8 \log{\left(\tanh{\left(\frac{1}{2} \right)} + 1 \right)} \tanh^{2}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} + \frac{2}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} - 2 i \pi - \frac{4 \left(\log{\left(- \tanh^{2}{\left(\frac{1}{2} \right)} + 2 \tanh{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right) \tanh^{2}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \left(\log{\left(- \tanh^{2}{\left(\frac{1}{2} \right)} + 2 \tanh{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right) \tanh^{4}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \left(\log{\left(- \tanh^{2}{\left(\frac{1}{2} \right)} + 2 \tanh{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right)}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1}$$
2/(1 + tanh(1/2)^4 - 2*tanh(1/2)^2) - 4*log(1 + tanh(1/2))/(1 + tanh(1/2)^4 - 2*tanh(1/2)^2) - 2*pi*i - 2*tanh(1/2)^2/(1 + tanh(1/2)^4 - 2*tanh(1/2)^2) - 2*tanh(1/2)/(1 + tanh(1/2)^4 - 2*tanh(1/2)^2) + 2*tanh(1/2)^3/(1 + tanh(1/2)^4 - 2*tanh(1/2)^2) + 2*tanh(1/2)^4/(1 + tanh(1/2)^4 - 2*tanh(1/2)^2) + 2*(pi*i + log(1 - tanh(1/2)^2 + 2*tanh(1/2)))/(1 + tanh(1/2)^4 - 2*tanh(1/2)^2) - 4*tanh(1/2)^2*(pi*i + log(1 - tanh(1/2)^2 + 2*tanh(1/2)))/(1 + tanh(1/2)^4 - 2*tanh(1/2)^2) - 4*tanh(1/2)^4*log(1 + tanh(1/2))/(1 + tanh(1/2)^4 - 2*tanh(1/2)^2) + 2*tanh(1/2)^4*(pi*i + log(1 - tanh(1/2)^2 + 2*tanh(1/2)))/(1 + tanh(1/2)^4 - 2*tanh(1/2)^2) + 8*tanh(1/2)^2*log(1 + tanh(1/2))/(1 + tanh(1/2)^4 - 2*tanh(1/2)^2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.