Sr Examen
Integral de sqrt(x^2-a^2)/x dx
Solución
Ha introducido
[src]
1 / | | _________ | / 2 2 | \/ x - a | ------------ dx | x | / 0
$$\int\limits_{0}^{1} \frac{\sqrt{- a^{2} + x^{2}}}{x}\, dx$$
Integral(sqrt(x^2 - a^2)/x, (x, 0, 1))
Respuesta (Indefinida)
[src]
// 2 | 2| \
|| /a\ I*x I*a |a | |
||- I*a*acosh|-| - --------------- + ----------------- for |--| > 1|
|| \x/ _________ _________ | 2| |
/ || / 2 / 2 |x | |
| || / a / a |
| _________ || / -1 + -- x* / -1 + -- |
| / 2 2 || / 2 / 2 |
| \/ x - a || \/ x \/ x |
| ------------ dx = C + |< |
| x || 2 |
| || /a\ x a |
/ || a*asin|-| + -------------- - ---------------- otherwise |
|| \x/ ________ ________ |
|| / 2 / 2 |
|| / a / a |
|| / 1 - -- x* / 1 - -- |
|| / 2 / 2 |
\\ \/ x \/ x /
$$\int \frac{\sqrt{- a^{2} + x^{2}}}{x}\, dx = C + \begin{cases} \frac{i a^{2}}{x \sqrt{\frac{a^{2}}{x^{2}} - 1}} - i a \operatorname{acosh}{\left(\frac{a}{x} \right)} - \frac{i x}{\sqrt{\frac{a^{2}}{x^{2}} - 1}} & \text{for}\: \left|{\frac{a^{2}}{x^{2}}}\right| > 1 \\- \frac{a^{2}}{x \sqrt{- \frac{a^{2}}{x^{2}} + 1}} + a \operatorname{asin}{\left(\frac{a}{x} \right)} + \frac{x}{\sqrt{- \frac{a^{2}}{x^{2}} + 1}} & \text{otherwise} \end{cases}$$
Respuesta
[src]
1 / | | / 3 2 2 | 2| | | I*x 2*I*x I*x*a I*a |a | | |- ------------ - ------------ + ------------ + ------------------ for ---- > 1 | | 3/2 _________ 3/2 _________ 2 | | / 2 2\ / 2 2 / 2 2\ / 2 x | | \a - x / \/ a - x \a - x / 2 / a | | x * / -1 + -- | | / 2 | | \/ x | < dx | | 3 2 2 | | x 2*x x*a a | |- ------------ + ------------ + ------------ - ----------------- otherwise | | 3/2 _________ 3/2 ________ | | / 2 2\ / 2 2 / 2 2\ / 2 | | \x - a / \/ x - a \x - a / 2 / a | | x * / 1 - -- | | / 2 | \ \/ x | / 0
$$\int\limits_{0}^{1} \begin{cases} \frac{i a^{2} x}{\left(a^{2} - x^{2}\right)^{\frac{3}{2}}} + \frac{i a^{2}}{x^{2} \sqrt{\frac{a^{2}}{x^{2}} - 1}} - \frac{i x^{3}}{\left(a^{2} - x^{2}\right)^{\frac{3}{2}}} - \frac{2 i x}{\sqrt{a^{2} - x^{2}}} & \text{for}\: \frac{\left|{a^{2}}\right|}{x^{2}} > 1 \\\frac{a^{2} x}{\left(- a^{2} + x^{2}\right)^{\frac{3}{2}}} - \frac{a^{2}}{x^{2} \sqrt{- \frac{a^{2}}{x^{2}} + 1}} - \frac{x^{3}}{\left(- a^{2} + x^{2}\right)^{\frac{3}{2}}} + \frac{2 x}{\sqrt{- a^{2} + x^{2}}} & \text{otherwise} \end{cases}\, dx$$
=
=
1 / | | / 3 2 2 | 2| | | I*x 2*I*x I*x*a I*a |a | | |- ------------ - ------------ + ------------ + ------------------ for ---- > 1 | | 3/2 _________ 3/2 _________ 2 | | / 2 2\ / 2 2 / 2 2\ / 2 x | | \a - x / \/ a - x \a - x / 2 / a | | x * / -1 + -- | | / 2 | | \/ x | < dx | | 3 2 2 | | x 2*x x*a a | |- ------------ + ------------ + ------------ - ----------------- otherwise | | 3/2 _________ 3/2 ________ | | / 2 2\ / 2 2 / 2 2\ / 2 | | \x - a / \/ x - a \x - a / 2 / a | | x * / 1 - -- | | / 2 | \ \/ x | / 0
$$\int\limits_{0}^{1} \begin{cases} \frac{i a^{2} x}{\left(a^{2} - x^{2}\right)^{\frac{3}{2}}} + \frac{i a^{2}}{x^{2} \sqrt{\frac{a^{2}}{x^{2}} - 1}} - \frac{i x^{3}}{\left(a^{2} - x^{2}\right)^{\frac{3}{2}}} - \frac{2 i x}{\sqrt{a^{2} - x^{2}}} & \text{for}\: \frac{\left|{a^{2}}\right|}{x^{2}} > 1 \\\frac{a^{2} x}{\left(- a^{2} + x^{2}\right)^{\frac{3}{2}}} - \frac{a^{2}}{x^{2} \sqrt{- \frac{a^{2}}{x^{2}} + 1}} - \frac{x^{3}}{\left(- a^{2} + x^{2}\right)^{\frac{3}{2}}} + \frac{2 x}{\sqrt{- a^{2} + x^{2}}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((-i*x^3/(a^2 - x^2)^(3/2) - 2*i*x/sqrt(a^2 - x^2) + i*x*a^2/(a^2 - x^2)^(3/2) + i*a^2/(x^2*sqrt(-1 + a^2/x^2)), |a^2|/x^2 > 1), (-x^3/(x^2 - a^2)^(3/2) + 2*x/sqrt(x^2 - a^2) + x*a^2/(x^2 - a^2)^(3/2) - a^2/(x^2*sqrt(1 - a^2/x^2)), True)), (x, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.