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