Sr Examen
Integral de 1/(sin(x/sqrt(2))) dx
Solución
Ha introducido
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1 / | | 1 | ---------- dx | / x \ | sin|-----| | | ___| | \\/ 2 / | / 0
$$\int\limits_{0}^{1} \frac{1}{\sin{\left(\frac{x}{\sqrt{2}} \right)}}\, dx$$
Integral(1/sin(x/sqrt(2)), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 1 | ---------- dx | / x \ | sin|-----| | | ___| | \\/ 2 / | /
La función subintegral
1 ---------- / x \ sin|-----| | ___| \\/ 2 /
Multiplicamos numerador y denominador por
/ ___\ |x*\/ 2 | sin|-------| \ 2 /
obtendremos
/ ___ \
|\/ 2 |
sin|-----*x|
1 \ 2 /
---------- = -------------
/ x \ / ___ \
sin|-----| 2|\/ 2 |
| ___| sin |-----*x|
\\/ 2 / \ 2 /Como
sin(a)^2 + cos(a)^2 = 1
entonces
/ ___\ / ___\
2|x*\/ 2 | 2|x*\/ 2 |
sin |-------| = 1 - cos |-------|
\ 2 / \ 2 /cambiamos denominador
/ ___ \ / ___ \
|\/ 2 | |\/ 2 |
sin|-----*x| sin|-----*x|
\ 2 / \ 2 /
------------- = -----------------
/ ___ \ / ___ \
2|\/ 2 | 2|\/ 2 |
sin |-----*x| 1 - cos |-----*x|
\ 2 / \ 2 /hacemos el cambio
/ ___\
|x*\/ 2 |
u = cos|-------|
\ 2 /entonces integral
/ | | / ___ \ | |\/ 2 | | sin|-----*x| | \ 2 / | ----------------- dx = | / ___ \ | 2|\/ 2 | | 1 - cos |-----*x| | \ 2 / | /
/ | | / ___ \ | |\/ 2 | | sin|-----*x| | \ 2 / | ----------------- dx = | / ___ \ | 2|\/ 2 | | 1 - cos |-----*x| | \ 2 / | /
Como du = -sqrt(2)*dx*sin(sqrt(2)*x/2)/2
/ | | / ___ \ | |\/ 2 | | sin|-----*x| | \ 2 / | ----------------- du | / ___ \ | 2|\/ 2 | | 1 - cos |-----*x| | \ 2 / | /
Reescribimos la función subintegral
/ ___ \
|\/ 2 |
sin|-----*x| ___
\ 2 / -\/ 2 / 1 1 \
----------------- = -------*|----- + -----|
/ ___ \ 2 \1 - u 1 + u/
2|\/ 2 |
1 - cos |-----*x|
\ 2 / entonces
/ / / \ / | | | | | ___ | | 1 | 1 | | / ___ \ -\/ 2 *| | ----- du + | ----- du| | |\/ 2 | | | 1 + u | 1 - u | | sin|-----*x| | | | | | \ 2 / \/ / / | ----------------- du = ----------------------------------- = | / ___ \ 2 | 2|\/ 2 | | 1 - cos |-----*x| | \ 2 / | /
= u*sin(x*sqrt(2)/2)/(1 - cos(x*sqrt(2)/2)^2)
hacemos cambio inverso
/ ___\
|x*\/ 2 |
u = cos|-------|
\ 2 /Respuesta
/ / / ___\\ / / ___\\\ / | | |x*\/ 2 || | |x*\/ 2 ||| | |log|-1 + cos|-------|| log|1 + cos|-------||| | 1 ___ | \ \ 2 // \ \ 2 //| | ---------- dx = \/ 2 *|---------------------- - ---------------------| | / x \ \ 2 2 / + C0 | sin|-----| | | ___| | \\/ 2 / | /
donde C0 es la constante que no depende de x
Respuesta (Indefinida)
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/ / / ___\\ / / ___\\\ / | | |x*\/ 2 || | |x*\/ 2 ||| | |log|-1 + cos|-------|| log|1 + cos|-------||| | 1 ___ | \ \ 2 // \ \ 2 //| | ---------- dx = C + \/ 2 *|---------------------- - ---------------------| | / x \ \ 2 2 / | sin|-----| | | ___| | \\/ 2 / | /
$$\int \frac{1}{\sin{\left(\frac{x}{\sqrt{2}} \right)}}\, dx = C + \sqrt{2} \left(\frac{\log{\left(\cos{\left(\frac{\sqrt{2} x}{2} \right)} - 1 \right)}}{2} - \frac{\log{\left(\cos{\left(\frac{\sqrt{2} x}{2} \right)} + 1 \right)}}{2}\right)$$
Gráfica
Respuesta
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/ / / ___\\ / / ___\\ \
| | |\/ 2 || | |\/ 2 || |
|log|1 - cos|-----|| log|1 + cos|-----|| |
___ | \ \ 2 // \ \ 2 // pi*I|
oo + \/ 2 *|------------------- - ------------------- + ----|
\ 2 2 2 /
$$\infty + \sqrt{2} \left(\frac{\log{\left(1 - \cos{\left(\frac{\sqrt{2}}{2} \right)} \right)}}{2} - \frac{\log{\left(\cos{\left(\frac{\sqrt{2}}{2} \right)} + 1 \right)}}{2} + \frac{i \pi}{2}\right)$$
=
=
/ / / ___\\ / / ___\\ \
| | |\/ 2 || | |\/ 2 || |
|log|1 - cos|-----|| log|1 + cos|-----|| |
___ | \ \ 2 // \ \ 2 // pi*I|
oo + \/ 2 *|------------------- - ------------------- + ----|
\ 2 2 2 /
$$\infty + \sqrt{2} \left(\frac{\log{\left(1 - \cos{\left(\frac{\sqrt{2}}{2} \right)} \right)}}{2} - \frac{\log{\left(\cos{\left(\frac{\sqrt{2}}{2} \right)} + 1 \right)}}{2} + \frac{i \pi}{2}\right)$$
oo + sqrt(2)*(log(1 - cos(sqrt(2)/2))/2 - log(1 + cos(sqrt(2)/2))/2 + pi*i/2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.