Sr Examen
Integral de 1/(x*sqrt(1-i*n^2*x)) dx
Solución
Ha introducido
[src]
___
\/ 2
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2
e
/
|
| 1
| ----------------- dx
| ____________
| / 2
| x*\/ 1 - I*n *x
|
/
1/2
e
$$\int\limits_{e^{\frac{1}{2}}}^{e^{\frac{\sqrt{2}}{2}}} \frac{1}{x \sqrt{- x i n^{2} + 1}}\, dx$$
Integral(1/(x*sqrt(1 - i*n^2*x)), (x, exp(1/2), exp(sqrt(2)/2)))
Respuesta (Indefinida)
[src]
// / -pi*I \ \
|| | ------| |
|| | 4 | |
|| |e | 1 |
/ ||-2*acosh|-------| for ------ > 1|
| || | ___| | 2| |
| 1 || \n*\/ x / |x*n | |
| ----------------- dx = C + |< |
| ____________ || / -pi*I \ |
| / 2 || | ------| |
| x*\/ 1 - I*n *x || | 4 | |
| || |e | |
/ ||2*I*asin|-------| otherwise |
|| | ___| |
\\ \n*\/ x / /
$$\int \frac{1}{x \sqrt{- x i n^{2} + 1}}\, dx = C + \begin{cases} - 2 \operatorname{acosh}{\left(\frac{e^{- \frac{i \pi}{4}}}{n \sqrt{x}} \right)} & \text{for}\: \frac{1}{\left|{n^{2} x}\right|} > 1 \\2 i \operatorname{asin}{\left(\frac{e^{- \frac{i \pi}{4}}}{n \sqrt{x}} \right)} & \text{otherwise} \end{cases}$$
Respuesta
[src]
___
\/ 2
-----
2
e
/
|
| / -pi*I
| | ------
| | 4
| | e 1
| |----------------------------- for ------ > 1
| | ______________ | 2|
| | / -pi*I x*|n |
| | / ------
| | / 2
| | 3/2 / e
| |n*x * / -1 + -------
| | / 2
| | \/ n *x
| < dx
| | -pi*I
| | ------
| | 4
| | -I*e
| |---------------------------- otherwise
| | _____________
| | / -pi*I
| | / ------
| | / 2
| | 3/2 / e
| |n*x * / 1 - -------
| | / 2
| \ \/ n *x
|
/
1/2
e
$$\int\limits_{e^{\frac{1}{2}}}^{e^{\frac{\sqrt{2}}{2}}} \begin{cases} \frac{e^{- \frac{i \pi}{4}}}{n x^{\frac{3}{2}} \sqrt{-1 + \frac{e^{- \frac{i \pi}{2}}}{n^{2} x}}} & \text{for}\: \frac{1}{x \left|{n^{2}}\right|} > 1 \\- \frac{i e^{- \frac{i \pi}{4}}}{n x^{\frac{3}{2}} \sqrt{1 - \frac{e^{- \frac{i \pi}{2}}}{n^{2} x}}} & \text{otherwise} \end{cases}\, dx$$
=
=
___
\/ 2
-----
2
e
/
|
| / -pi*I
| | ------
| | 4
| | e 1
| |----------------------------- for ------ > 1
| | ______________ | 2|
| | / -pi*I x*|n |
| | / ------
| | / 2
| | 3/2 / e
| |n*x * / -1 + -------
| | / 2
| | \/ n *x
| < dx
| | -pi*I
| | ------
| | 4
| | -I*e
| |---------------------------- otherwise
| | _____________
| | / -pi*I
| | / ------
| | / 2
| | 3/2 / e
| |n*x * / 1 - -------
| | / 2
| \ \/ n *x
|
/
1/2
e
$$\int\limits_{e^{\frac{1}{2}}}^{e^{\frac{\sqrt{2}}{2}}} \begin{cases} \frac{e^{- \frac{i \pi}{4}}}{n x^{\frac{3}{2}} \sqrt{-1 + \frac{e^{- \frac{i \pi}{2}}}{n^{2} x}}} & \text{for}\: \frac{1}{x \left|{n^{2}}\right|} > 1 \\- \frac{i e^{- \frac{i \pi}{4}}}{n x^{\frac{3}{2}} \sqrt{1 - \frac{e^{- \frac{i \pi}{2}}}{n^{2} x}}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((exp_polar(-pi*i/4)/(n*x^(3/2)*sqrt(-1 + exp_polar(-pi*i/2)/(n^2*x))), 1/(x*|n^2|) > 1), (-i*exp_polar(-pi*i/4)/(n*x^(3/2)*sqrt(1 - exp_polar(-pi*i/2)/(n^2*x))), True)), (x, exp(1/2), exp(sqrt(2)/2)))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.