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