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