Sr Examen
Integral de sin(k*Pi/x) dx
Solución
Ha introducido
[src]
-1 + k
/
|
| /k*pi\
| sin|----| dx
| \ x /
|
/
2
$$\int\limits_{2}^{k - 1} \sin{\left(\frac{\pi k}{x} \right)}\, dx$$
Integral(sin((k*pi)/x), (x, 2, -1 + k))
Respuesta (Indefinida)
[src]
/ 2\
|k |
/ pi*k*log|--|
| | 2|
| /k*pi\ /pi*k\ /k\ /pi*k\ \x /
| sin|----| dx = C + x*sin|----| + pi*k*log|-| - pi*k*Ci|----| - ------------
| \ x / \ x / \x/ \ x / 2
|
/
$$\int \sin{\left(\frac{\pi k}{x} \right)}\, dx = C + \pi k \log{\left(\frac{k}{x} \right)} - \frac{\pi k \log{\left(\frac{k^{2}}{x^{2}} \right)}}{2} - \pi k \operatorname{Ci}{\left(\frac{\pi k}{x} \right)} + x \sin{\left(\frac{\pi k}{x} \right)}$$
Respuesta
[src]
/ 2 \
/ 2\ | k |
|k | pi*k*log|---------|
pi*k*log|--| | 2|
/pi*k\ / pi*k \ /pi*k\ / k \ \4 / / pi*k \ /k\ \(-1 + k) /
- 2*sin|----| + (-1 + k)*sin|------| + pi*k*Ci|----| + pi*k*log|------| + ------------ - pi*k*Ci|------| - pi*k*log|-| - -------------------
\ 2 / \-1 + k/ \ 2 / \-1 + k/ 2 \-1 + k/ \2/ 2
$$- \pi k \log{\left(\frac{k}{2} \right)} + \frac{\pi k \log{\left(\frac{k^{2}}{4} \right)}}{2} + \pi k \log{\left(\frac{k}{k - 1} \right)} - \frac{\pi k \log{\left(\frac{k^{2}}{\left(k - 1\right)^{2}} \right)}}{2} + \pi k \operatorname{Ci}{\left(\frac{\pi k}{2} \right)} - \pi k \operatorname{Ci}{\left(\frac{\pi k}{k - 1} \right)} + \left(k - 1\right) \sin{\left(\frac{\pi k}{k - 1} \right)} - 2 \sin{\left(\frac{\pi k}{2} \right)}$$
=
=
/ 2 \
/ 2\ | k |
|k | pi*k*log|---------|
pi*k*log|--| | 2|
/pi*k\ / pi*k \ /pi*k\ / k \ \4 / / pi*k \ /k\ \(-1 + k) /
- 2*sin|----| + (-1 + k)*sin|------| + pi*k*Ci|----| + pi*k*log|------| + ------------ - pi*k*Ci|------| - pi*k*log|-| - -------------------
\ 2 / \-1 + k/ \ 2 / \-1 + k/ 2 \-1 + k/ \2/ 2
$$- \pi k \log{\left(\frac{k}{2} \right)} + \frac{\pi k \log{\left(\frac{k^{2}}{4} \right)}}{2} + \pi k \log{\left(\frac{k}{k - 1} \right)} - \frac{\pi k \log{\left(\frac{k^{2}}{\left(k - 1\right)^{2}} \right)}}{2} + \pi k \operatorname{Ci}{\left(\frac{\pi k}{2} \right)} - \pi k \operatorname{Ci}{\left(\frac{\pi k}{k - 1} \right)} + \left(k - 1\right) \sin{\left(\frac{\pi k}{k - 1} \right)} - 2 \sin{\left(\frac{\pi k}{2} \right)}$$
-2*sin(pi*k/2) + (-1 + k)*sin(pi*k/(-1 + k)) + pi*k*Ci(pi*k/2) + pi*k*log(k/(-1 + k)) + pi*k*log(k^2/4)/2 - pi*k*Ci(pi*k/(-1 + k)) - pi*k*log(k/2) - pi*k*log(k^2/(-1 + k)^2)/2
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.