Sr Examen
Integral de 1/((x-y)^2-R^2-i*e) dy
Solución
Ha introducido
[src]
1 / | | 1 | ------------------- dy | 2 2 | (x - y) - r - I*E | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(- r^{2} + \left(x - y\right)^{2}\right) - e i}\, dy$$
Integral(1/((x - y)^2 - r^2 - i*E), (y, 0, 1))
Respuesta (Indefinida)
[src]
/ y - x \
atan|---------------|
/ | ____________|
| | / 2 |
| 1 \\/ - r - E*I /
| ------------------- dy = C + ---------------------
| 2 2 ____________
| (x - y) - r - I*E / 2
| \/ - r - E*I
/
$$\int \frac{1}{\left(- r^{2} + \left(x - y\right)^{2}\right) - e i}\, dy = C + \frac{\operatorname{atan}{\left(\frac{- x + y}{\sqrt{- r^{2} - e i}} \right)}}{\sqrt{- r^{2} - e i}}$$
Respuesta
[src]
/ __________ \ / __________ \ / __________ \ / __________ \
| / 1 / 2 \| | / 1 / 2 \| | / 1 / 2 \| | / 1 / 2 \|
| / -------- *\- 2*r - 2*E*I/| | / -------- *\- 2*r - 2*E*I/| | / -------- *\- 2*r - 2*E*I/| | / -------- *\- 2*r - 2*E*I/|
__________ | / 2 | __________ | / 2 | __________ | / 2 | __________ | / 2 |
/ 1 | \/ r + E*I | / 1 | \/ r + E*I | / 1 | \/ r + E*I | / 1 | \/ r + E*I |
/ -------- *log|-x - --------------------------------| / -------- *log|1 - x + --------------------------------| / -------- *log|-x + --------------------------------| / -------- *log|1 - x - --------------------------------|
/ 2 \ 2 / / 2 \ 2 / / 2 \ 2 / / 2 \ 2 /
\/ r + E*I \/ r + E*I \/ r + E*I \/ r + E*I
---------------------------------------------------------- + ------------------------------------------------------------- - ---------------------------------------------------------- - -------------------------------------------------------------
2 2 2 2
$$\frac{\sqrt{\frac{1}{r^{2} + e i}} \log{\left(- x - \frac{\left(- 2 r^{2} - 2 e i\right) \sqrt{\frac{1}{r^{2} + e i}}}{2} \right)}}{2} - \frac{\sqrt{\frac{1}{r^{2} + e i}} \log{\left(- x + \frac{\left(- 2 r^{2} - 2 e i\right) \sqrt{\frac{1}{r^{2} + e i}}}{2} \right)}}{2} - \frac{\sqrt{\frac{1}{r^{2} + e i}} \log{\left(- x - \frac{\left(- 2 r^{2} - 2 e i\right) \sqrt{\frac{1}{r^{2} + e i}}}{2} + 1 \right)}}{2} + \frac{\sqrt{\frac{1}{r^{2} + e i}} \log{\left(- x + \frac{\left(- 2 r^{2} - 2 e i\right) \sqrt{\frac{1}{r^{2} + e i}}}{2} + 1 \right)}}{2}$$
=
=
/ __________ \ / __________ \ / __________ \ / __________ \
| / 1 / 2 \| | / 1 / 2 \| | / 1 / 2 \| | / 1 / 2 \|
| / -------- *\- 2*r - 2*E*I/| | / -------- *\- 2*r - 2*E*I/| | / -------- *\- 2*r - 2*E*I/| | / -------- *\- 2*r - 2*E*I/|
__________ | / 2 | __________ | / 2 | __________ | / 2 | __________ | / 2 |
/ 1 | \/ r + E*I | / 1 | \/ r + E*I | / 1 | \/ r + E*I | / 1 | \/ r + E*I |
/ -------- *log|-x - --------------------------------| / -------- *log|1 - x + --------------------------------| / -------- *log|-x + --------------------------------| / -------- *log|1 - x - --------------------------------|
/ 2 \ 2 / / 2 \ 2 / / 2 \ 2 / / 2 \ 2 /
\/ r + E*I \/ r + E*I \/ r + E*I \/ r + E*I
---------------------------------------------------------- + ------------------------------------------------------------- - ---------------------------------------------------------- - -------------------------------------------------------------
2 2 2 2
$$\frac{\sqrt{\frac{1}{r^{2} + e i}} \log{\left(- x - \frac{\left(- 2 r^{2} - 2 e i\right) \sqrt{\frac{1}{r^{2} + e i}}}{2} \right)}}{2} - \frac{\sqrt{\frac{1}{r^{2} + e i}} \log{\left(- x + \frac{\left(- 2 r^{2} - 2 e i\right) \sqrt{\frac{1}{r^{2} + e i}}}{2} \right)}}{2} - \frac{\sqrt{\frac{1}{r^{2} + e i}} \log{\left(- x - \frac{\left(- 2 r^{2} - 2 e i\right) \sqrt{\frac{1}{r^{2} + e i}}}{2} + 1 \right)}}{2} + \frac{\sqrt{\frac{1}{r^{2} + e i}} \log{\left(- x + \frac{\left(- 2 r^{2} - 2 e i\right) \sqrt{\frac{1}{r^{2} + e i}}}{2} + 1 \right)}}{2}$$
sqrt(1/(r^2 + E*i))*log(-x - sqrt(1/(r^2 + E*i))*(-2*r^2 - 2*E*i)/2)/2 + sqrt(1/(r^2 + E*i))*log(1 - x + sqrt(1/(r^2 + E*i))*(-2*r^2 - 2*E*i)/2)/2 - sqrt(1/(r^2 + E*i))*log(-x + sqrt(1/(r^2 + E*i))*(-2*r^2 - 2*E*i)/2)/2 - sqrt(1/(r^2 + E*i))*log(1 - x - sqrt(1/(r^2 + E*i))*(-2*r^2 - 2*E*i)/2)/2
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.