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