Sr Examen
Integral de sqrt(a^2-x^2) dx
Solución
Ha introducido
[src]
a / | | _________ | / 2 2 | \/ a - x dx | / 0
$$\int\limits_{0}^{a} \sqrt{a^{2} - x^{2}}\, dx$$
Integral(sqrt(a^2 - x^2), (x, 0, a))
Respuesta (Indefinida)
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// 2 /x\ \
|| I*a *acosh|-| 3 | 2| |
|| \a/ I*x I*a*x |x | |
||- ------------- + ------------------- - ----------------- for |--| > 1|
|| 2 _________ _________ | 2| |
/ || / 2 / 2 |a | |
| || / x / x |
| _________ || 2*a* / -1 + -- 2* / -1 + -- |
| / 2 2 || / 2 / 2 |
| \/ a - x dx = C + |< \/ a \/ a |
| || |
/ || ________ |
|| / 2 |
|| / x |
|| 2 /x\ a*x* / 1 - -- |
|| a *asin|-| / 2 |
|| \a/ \/ a |
|| ---------- + ------------------ otherwise |
\\ 2 2 /
$$\int \sqrt{a^{2} - x^{2}}\, dx = C + \begin{cases} - \frac{i a^{2} \operatorname{acosh}{\left(\frac{x}{a} \right)}}{2} - \frac{i a x}{2 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{i x^{3}}{2 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} & \text{for}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{a^{2} \operatorname{asin}{\left(\frac{x}{a} \right)}}{2} + \frac{a x \sqrt{1 - \frac{x^{2}}{a^{2}}}}{2} & \text{otherwise} \end{cases}$$
Respuesta
[src]
/ 0 | / | | | | / 2 4 2 | 2| | | | I*a I*x I*x 3*I*x |x | | | |- --------------- + ---------------- - ----------------- + ------------------- for ---- > 1 | | | _________ 3/2 3/2 _________ 2 | | | / 2 / 2\ / 2\ / 2 a | | | / x | x | 3 | x | / x | | | / -1 + -- 2*a*|-1 + --| 2*a *|-1 + --| 2*a* / -1 + -- | | | / 2 | 2| | 2| / 2 | | | \/ a \ a / \ a / \/ a | | | | | | ________ |- | < / 2 dx for a < 0 | | | / x | | | a* / 1 - -- | | | / 2 2 | | | \/ a a x | | | ---------------- + ---------------- - ------------------ otherwise | | | 2 ________ ________ | | | / 2 / 2 | | | / x / x | | | 2* / 1 - -- 2*a* / 1 - -- | | | / 2 / 2 | | \ \/ a \/ a | | | / | a < | a | / | | | | / 2 4 2 | 2| | | | I*a I*x I*x 3*I*x |x | | | |- --------------- + ---------------- - ----------------- + ------------------- for ---- > 1 | | | _________ 3/2 3/2 _________ 2 | | | / 2 / 2\ / 2\ / 2 a | | | / x | x | 3 | x | / x | | | / -1 + -- 2*a*|-1 + --| 2*a *|-1 + --| 2*a* / -1 + -- | | | / 2 | 2| | 2| / 2 | | | \/ a \ a / \ a / \/ a | | | | | | ________ | | < / 2 dx otherwise | | | / x | | | a* / 1 - -- | | | / 2 2 | | | \/ a a x | | | ---------------- + ---------------- - ------------------ otherwise | | | 2 ________ ________ | | | / 2 / 2 | | | / x / x | | | 2* / 1 - -- 2*a* / 1 - -- | | | / 2 / 2 | | \ \/ a \/ a | | |/ \0
$$\begin{cases} - \int\limits_{a}^{0} \begin{cases} - \frac{i a}{\sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{3 i x^{2}}{2 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{i x^{2}}{2 a \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{i x^{4}}{2 a^{3} \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{\left|{x^{2}}\right|}{a^{2}} > 1 \\\frac{a \sqrt{1 - \frac{x^{2}}{a^{2}}}}{2} + \frac{a}{2 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{x^{2}}{2 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\, dx & \text{for}\: a < 0 \\\int\limits_{0}^{a} \begin{cases} - \frac{i a}{\sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{3 i x^{2}}{2 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{i x^{2}}{2 a \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{i x^{4}}{2 a^{3} \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{\left|{x^{2}}\right|}{a^{2}} > 1 \\\frac{a \sqrt{1 - \frac{x^{2}}{a^{2}}}}{2} + \frac{a}{2 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{x^{2}}{2 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases}$$
=
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/ 0 | / | | | | / 2 4 2 | 2| | | | I*a I*x I*x 3*I*x |x | | | |- --------------- + ---------------- - ----------------- + ------------------- for ---- > 1 | | | _________ 3/2 3/2 _________ 2 | | | / 2 / 2\ / 2\ / 2 a | | | / x | x | 3 | x | / x | | | / -1 + -- 2*a*|-1 + --| 2*a *|-1 + --| 2*a* / -1 + -- | | | / 2 | 2| | 2| / 2 | | | \/ a \ a / \ a / \/ a | | | | | | ________ |- | < / 2 dx for a < 0 | | | / x | | | a* / 1 - -- | | | / 2 2 | | | \/ a a x | | | ---------------- + ---------------- - ------------------ otherwise | | | 2 ________ ________ | | | / 2 / 2 | | | / x / x | | | 2* / 1 - -- 2*a* / 1 - -- | | | / 2 / 2 | | \ \/ a \/ a | | | / | a < | a | / | | | | / 2 4 2 | 2| | | | I*a I*x I*x 3*I*x |x | | | |- --------------- + ---------------- - ----------------- + ------------------- for ---- > 1 | | | _________ 3/2 3/2 _________ 2 | | | / 2 / 2\ / 2\ / 2 a | | | / x | x | 3 | x | / x | | | / -1 + -- 2*a*|-1 + --| 2*a *|-1 + --| 2*a* / -1 + -- | | | / 2 | 2| | 2| / 2 | | | \/ a \ a / \ a / \/ a | | | | | | ________ | | < / 2 dx otherwise | | | / x | | | a* / 1 - -- | | | / 2 2 | | | \/ a a x | | | ---------------- + ---------------- - ------------------ otherwise | | | 2 ________ ________ | | | / 2 / 2 | | | / x / x | | | 2* / 1 - -- 2*a* / 1 - -- | | | / 2 / 2 | | \ \/ a \/ a | | |/ \0
$$\begin{cases} - \int\limits_{a}^{0} \begin{cases} - \frac{i a}{\sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{3 i x^{2}}{2 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{i x^{2}}{2 a \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{i x^{4}}{2 a^{3} \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{\left|{x^{2}}\right|}{a^{2}} > 1 \\\frac{a \sqrt{1 - \frac{x^{2}}{a^{2}}}}{2} + \frac{a}{2 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{x^{2}}{2 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\, dx & \text{for}\: a < 0 \\\int\limits_{0}^{a} \begin{cases} - \frac{i a}{\sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{3 i x^{2}}{2 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{i x^{2}}{2 a \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{i x^{4}}{2 a^{3} \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{\left|{x^{2}}\right|}{a^{2}} > 1 \\\frac{a \sqrt{1 - \frac{x^{2}}{a^{2}}}}{2} + \frac{a}{2 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{x^{2}}{2 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases}$$
Piecewise((-Integral(Piecewise((-i*a/sqrt(-1 + x^2/a^2) + i*x^2/(2*a*(-1 + x^2/a^2)^(3/2)) - i*x^4/(2*a^3*(-1 + x^2/a^2)^(3/2)) + 3*i*x^2/(2*a*sqrt(-1 + x^2/a^2)), |x^2|/a^2 > 1), (a*sqrt(1 - x^2/a^2)/2 + a/(2*sqrt(1 - x^2/a^2)) - x^2/(2*a*sqrt(1 - x^2/a^2)), True)), (x, a, 0)), a < 0), (Integral(Piecewise((-i*a/sqrt(-1 + x^2/a^2) + i*x^2/(2*a*(-1 + x^2/a^2)^(3/2)) - i*x^4/(2*a^3*(-1 + x^2/a^2)^(3/2)) + 3*i*x^2/(2*a*sqrt(-1 + x^2/a^2)), |x^2|/a^2 > 1), (a*sqrt(1 - x^2/a^2)/2 + a/(2*sqrt(1 - x^2/a^2)) - x^2/(2*a*sqrt(1 - x^2/a^2)), True)), (x, 0, a)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.