Sr Examen
Integral de (r^2-y^2)^(3/2) dy
Solución
Ha introducido
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-r*\/ 2
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$$\int\limits_{- \frac{\sqrt{2} r}{2}}^{0} \left(r^{2} - y^{2}\right)^{\frac{3}{2}}\, dy$$
Integral((r^2 - y^2)^(3/2), (y, -r*sqrt(2)/2, 0))
Solución detallada
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Vuelva a escribir el integrando:
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Integramos término a término:
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La integral del producto de una función por una constante es la constante por la integral de esta función:
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No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
Por lo tanto, el resultado es:
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La integral del producto de una función por una constante es la constante por la integral de esta función:
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No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
Por lo tanto, el resultado es:
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El resultado es:
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Ahora simplificar:
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Añadimos la constante de integración:
Respuesta:
Respuesta (Indefinida)
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// 4 /y\ \
|| I*r *acosh|-| 3 5 3 | 2| | // 2 /y\ \
|| \r/ 3*I*r*y I*y I*y*r |y | | || I*r *acosh|-| 3 | 2| |
||- ------------- - ----------------- + ------------------- + ----------------- for |--| > 1| || \r/ I*y I*r*y |y | |
|| 8 _________ _________ _________ | 2| | ||- ------------- + ------------------- - ----------------- for |--| > 1|
|| / 2 / 2 / 2 |r | | || 2 _________ _________ | 2| |
/ || / y / y / y | || / 2 / 2 |r | |
| || 8* / -1 + -- 4*r* / -1 + -- 8* / -1 + -- | || / y / y |
| 3/2 || / 2 / 2 / 2 | || 2*r* / -1 + -- 2* / -1 + -- |
| / 2 2\ || \/ r \/ r \/ r | 2 || / 2 / 2 |
| \r - y / dy = C - |< | + r *|< \/ r \/ r |
| || 4 /y\ | || |
/ || r *asin|-| 5 3 3 | || ________ |
|| \r/ y y*r 3*r*y | || / 2 |
|| ---------- - ------------------ - ---------------- + ---------------- otherwise | || / y |
|| 8 ________ ________ ________ | || 2 /y\ r*y* / 1 - -- |
|| / 2 / 2 / 2 | || r *asin|-| / 2 |
|| / y / y / y | || \r/ \/ r |
|| 4*r* / 1 - -- 8* / 1 - -- 8* / 1 - -- | || ---------- + ------------------ otherwise |
|| / 2 / 2 / 2 | \\ 2 2 /
\\ \/ r \/ r \/ r /
$$\int \left(r^{2} - y^{2}\right)^{\frac{3}{2}}\, dy = C + r^{2} \left(\begin{cases} - \frac{i r^{2} \operatorname{acosh}{\left(\frac{y}{r} \right)}}{2} - \frac{i r y}{2 \sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{i y^{3}}{2 r \sqrt{-1 + \frac{y^{2}}{r^{2}}}} & \text{for}\: \left|{\frac{y^{2}}{r^{2}}}\right| > 1 \\\frac{r^{2} \operatorname{asin}{\left(\frac{y}{r} \right)}}{2} + \frac{r y \sqrt{1 - \frac{y^{2}}{r^{2}}}}{2} & \text{otherwise} \end{cases}\right) - \begin{cases} - \frac{i r^{4} \operatorname{acosh}{\left(\frac{y}{r} \right)}}{8} + \frac{i r^{3} y}{8 \sqrt{-1 + \frac{y^{2}}{r^{2}}}} - \frac{3 i r y^{3}}{8 \sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{i y^{5}}{4 r \sqrt{-1 + \frac{y^{2}}{r^{2}}}} & \text{for}\: \left|{\frac{y^{2}}{r^{2}}}\right| > 1 \\\frac{r^{4} \operatorname{asin}{\left(\frac{y}{r} \right)}}{8} - \frac{r^{3} y}{8 \sqrt{1 - \frac{y^{2}}{r^{2}}}} + \frac{3 r y^{3}}{8 \sqrt{1 - \frac{y^{2}}{r^{2}}}} - \frac{y^{5}}{4 r \sqrt{1 - \frac{y^{2}}{r^{2}}}} & \text{otherwise} \end{cases}$$
Respuesta
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/ 0 | / | | | | / 3 4 4 6 2 2 | 2| | | | I*r 7*I*y 5*I*y I*y 5*I*r*y 21*I*r*y |y | | | |- --------------- - ---------------- - ------------------- + ----------------- + -------------- + ----------------- for |--| > 1 | | | _________ 3/2 _________ 3/2 3/2 _________ | 2| | | | / 2 / 2\ / 2 / 2\ / 2\ / 2 |r | | | | / y | y | / y 3 | y | | y | / y | | | / -1 + -- 8*r*|-1 + --| 4*r* / -1 + -- 4*r *|-1 + --| 8*|-1 + --| 8* / -1 + -- | | | / 2 | 2| / 2 | 2| | 2| / 2 | | | \/ r \ r / \/ r \ r / \ r / \/ r | | | ___ | | | ________ ________ r*\/ 2 | | < / 2 / 2 dy for ------- > 0 | | | 3 / y 2 / y 2 | | | 5*r * / 1 - -- 3*r*y * / 1 - -- | | | 3 / 2 2 / 2 4 | | | 3*r \/ r 5*r*y \/ r y | | | ---------------- + ------------------- - ---------------- - --------------------- + ------------------ otherwise | | | ________ 8 ________ 4 ________ | | | / 2 / 2 / 2 | | | / y / y / y | | | 8* / 1 - -- 8* / 1 - -- 4*r* / 1 - -- | | | / 2 / 2 / 2 | | \ \/ r \/ r \/ r | | | / | ___ |-r*\/ 2 |--------- | 2 < | ___ | -r*\/ 2 | --------- | 2 | / | | | | / 3 4 4 6 2 2 | 2| | | | I*r 7*I*y 5*I*y I*y 5*I*r*y 21*I*r*y |y | | | |- --------------- - ---------------- - ------------------- + ----------------- + -------------- + ----------------- for |--| > 1 | | | _________ 3/2 _________ 3/2 3/2 _________ | 2| | | | / 2 / 2\ / 2 / 2\ / 2\ / 2 |r | | | | / y | y | / y 3 | y | | y | / y | | | / -1 + -- 8*r*|-1 + --| 4*r* / -1 + -- 4*r *|-1 + --| 8*|-1 + --| 8* / -1 + -- | | | / 2 | 2| / 2 | 2| | 2| / 2 | | | \/ r \ r / \/ r \ r / \ r / \/ r | | | | | | ________ ________ |- | < / 2 / 2 dy otherwise | | | 3 / y 2 / y | | | 5*r * / 1 - -- 3*r*y * / 1 - -- | | | 3 / 2 2 / 2 4 | | | 3*r \/ r 5*r*y \/ r y | | | ---------------- + ------------------- - ---------------- - --------------------- + ------------------ otherwise | | | ________ 8 ________ 4 ________ | | | / 2 / 2 / 2 | | | / y / y / y | | | 8* / 1 - -- 8* / 1 - -- 4*r* / 1 - -- | | | / 2 / 2 / 2 | | \ \/ r \/ r \/ r | | | / \ 0
$$\begin{cases} \int\limits_{- \frac{\sqrt{2} r}{2}}^{0} \begin{cases} - \frac{i r^{3}}{\sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{21 i r y^{2}}{8 \sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{5 i r y^{2}}{8 \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} - \frac{5 i y^{4}}{4 r \sqrt{-1 + \frac{y^{2}}{r^{2}}}} - \frac{7 i y^{4}}{8 r \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} + \frac{i y^{6}}{4 r^{3} \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \left|{\frac{y^{2}}{r^{2}}}\right| > 1 \\\frac{5 r^{3} \sqrt{1 - \frac{y^{2}}{r^{2}}}}{8} + \frac{3 r^{3}}{8 \sqrt{1 - \frac{y^{2}}{r^{2}}}} - \frac{3 r y^{2} \sqrt{1 - \frac{y^{2}}{r^{2}}}}{4} - \frac{5 r y^{2}}{8 \sqrt{1 - \frac{y^{2}}{r^{2}}}} + \frac{y^{4}}{4 r \sqrt{1 - \frac{y^{2}}{r^{2}}}} & \text{otherwise} \end{cases}\, dy & \text{for}\: \frac{\sqrt{2} r}{2} > 0 \\- \int\limits_{0}^{- \frac{\sqrt{2} r}{2}} \begin{cases} - \frac{i r^{3}}{\sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{21 i r y^{2}}{8 \sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{5 i r y^{2}}{8 \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} - \frac{5 i y^{4}}{4 r \sqrt{-1 + \frac{y^{2}}{r^{2}}}} - \frac{7 i y^{4}}{8 r \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} + \frac{i y^{6}}{4 r^{3} \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \left|{\frac{y^{2}}{r^{2}}}\right| > 1 \\\frac{5 r^{3} \sqrt{1 - \frac{y^{2}}{r^{2}}}}{8} + \frac{3 r^{3}}{8 \sqrt{1 - \frac{y^{2}}{r^{2}}}} - \frac{3 r y^{2} \sqrt{1 - \frac{y^{2}}{r^{2}}}}{4} - \frac{5 r y^{2}}{8 \sqrt{1 - \frac{y^{2}}{r^{2}}}} + \frac{y^{4}}{4 r \sqrt{1 - \frac{y^{2}}{r^{2}}}} & \text{otherwise} \end{cases}\, dy & \text{otherwise} \end{cases}$$
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/ 0 | / | | | | / 3 4 4 6 2 2 | 2| | | | I*r 7*I*y 5*I*y I*y 5*I*r*y 21*I*r*y |y | | | |- --------------- - ---------------- - ------------------- + ----------------- + -------------- + ----------------- for |--| > 1 | | | _________ 3/2 _________ 3/2 3/2 _________ | 2| | | | / 2 / 2\ / 2 / 2\ / 2\ / 2 |r | | | | / y | y | / y 3 | y | | y | / y | | | / -1 + -- 8*r*|-1 + --| 4*r* / -1 + -- 4*r *|-1 + --| 8*|-1 + --| 8* / -1 + -- | | | / 2 | 2| / 2 | 2| | 2| / 2 | | | \/ r \ r / \/ r \ r / \ r / \/ r | | | ___ | | | ________ ________ r*\/ 2 | | < / 2 / 2 dy for ------- > 0 | | | 3 / y 2 / y 2 | | | 5*r * / 1 - -- 3*r*y * / 1 - -- | | | 3 / 2 2 / 2 4 | | | 3*r \/ r 5*r*y \/ r y | | | ---------------- + ------------------- - ---------------- - --------------------- + ------------------ otherwise | | | ________ 8 ________ 4 ________ | | | / 2 / 2 / 2 | | | / y / y / y | | | 8* / 1 - -- 8* / 1 - -- 4*r* / 1 - -- | | | / 2 / 2 / 2 | | \ \/ r \/ r \/ r | | | / | ___ |-r*\/ 2 |--------- | 2 < | ___ | -r*\/ 2 | --------- | 2 | / | | | | / 3 4 4 6 2 2 | 2| | | | I*r 7*I*y 5*I*y I*y 5*I*r*y 21*I*r*y |y | | | |- --------------- - ---------------- - ------------------- + ----------------- + -------------- + ----------------- for |--| > 1 | | | _________ 3/2 _________ 3/2 3/2 _________ | 2| | | | / 2 / 2\ / 2 / 2\ / 2\ / 2 |r | | | | / y | y | / y 3 | y | | y | / y | | | / -1 + -- 8*r*|-1 + --| 4*r* / -1 + -- 4*r *|-1 + --| 8*|-1 + --| 8* / -1 + -- | | | / 2 | 2| / 2 | 2| | 2| / 2 | | | \/ r \ r / \/ r \ r / \ r / \/ r | | | | | | ________ ________ |- | < / 2 / 2 dy otherwise | | | 3 / y 2 / y | | | 5*r * / 1 - -- 3*r*y * / 1 - -- | | | 3 / 2 2 / 2 4 | | | 3*r \/ r 5*r*y \/ r y | | | ---------------- + ------------------- - ---------------- - --------------------- + ------------------ otherwise | | | ________ 8 ________ 4 ________ | | | / 2 / 2 / 2 | | | / y / y / y | | | 8* / 1 - -- 8* / 1 - -- 4*r* / 1 - -- | | | / 2 / 2 / 2 | | \ \/ r \/ r \/ r | | | / \ 0
$$\begin{cases} \int\limits_{- \frac{\sqrt{2} r}{2}}^{0} \begin{cases} - \frac{i r^{3}}{\sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{21 i r y^{2}}{8 \sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{5 i r y^{2}}{8 \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} - \frac{5 i y^{4}}{4 r \sqrt{-1 + \frac{y^{2}}{r^{2}}}} - \frac{7 i y^{4}}{8 r \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} + \frac{i y^{6}}{4 r^{3} \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \left|{\frac{y^{2}}{r^{2}}}\right| > 1 \\\frac{5 r^{3} \sqrt{1 - \frac{y^{2}}{r^{2}}}}{8} + \frac{3 r^{3}}{8 \sqrt{1 - \frac{y^{2}}{r^{2}}}} - \frac{3 r y^{2} \sqrt{1 - \frac{y^{2}}{r^{2}}}}{4} - \frac{5 r y^{2}}{8 \sqrt{1 - \frac{y^{2}}{r^{2}}}} + \frac{y^{4}}{4 r \sqrt{1 - \frac{y^{2}}{r^{2}}}} & \text{otherwise} \end{cases}\, dy & \text{for}\: \frac{\sqrt{2} r}{2} > 0 \\- \int\limits_{0}^{- \frac{\sqrt{2} r}{2}} \begin{cases} - \frac{i r^{3}}{\sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{21 i r y^{2}}{8 \sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{5 i r y^{2}}{8 \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} - \frac{5 i y^{4}}{4 r \sqrt{-1 + \frac{y^{2}}{r^{2}}}} - \frac{7 i y^{4}}{8 r \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} + \frac{i y^{6}}{4 r^{3} \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \left|{\frac{y^{2}}{r^{2}}}\right| > 1 \\\frac{5 r^{3} \sqrt{1 - \frac{y^{2}}{r^{2}}}}{8} + \frac{3 r^{3}}{8 \sqrt{1 - \frac{y^{2}}{r^{2}}}} - \frac{3 r y^{2} \sqrt{1 - \frac{y^{2}}{r^{2}}}}{4} - \frac{5 r y^{2}}{8 \sqrt{1 - \frac{y^{2}}{r^{2}}}} + \frac{y^{4}}{4 r \sqrt{1 - \frac{y^{2}}{r^{2}}}} & \text{otherwise} \end{cases}\, dy & \text{otherwise} \end{cases}$$
Piecewise((Integral(Piecewise((-i*r^3/sqrt(-1 + y^2/r^2) - 7*i*y^4/(8*r*(-1 + y^2/r^2)^(3/2)) - 5*i*y^4/(4*r*sqrt(-1 + y^2/r^2)) + i*y^6/(4*r^3*(-1 + y^2/r^2)^(3/2)) + 5*i*r*y^2/(8*(-1 + y^2/r^2)^(3/2)) + 21*i*r*y^2/(8*sqrt(-1 + y^2/r^2)), |y^2/r^2| > 1), (3*r^3/(8*sqrt(1 - y^2/r^2)) + 5*r^3*sqrt(1 - y^2/r^2)/8 - 5*r*y^2/(8*sqrt(1 - y^2/r^2)) - 3*r*y^2*sqrt(1 - y^2/r^2)/4 + y^4/(4*r*sqrt(1 - y^2/r^2)), True)), (y, -r*sqrt(2)/2, 0)), r*sqrt(2)/2 > 0), (-Integral(Piecewise((-i*r^3/sqrt(-1 + y^2/r^2) - 7*i*y^4/(8*r*(-1 + y^2/r^2)^(3/2)) - 5*i*y^4/(4*r*sqrt(-1 + y^2/r^2)) + i*y^6/(4*r^3*(-1 + y^2/r^2)^(3/2)) + 5*i*r*y^2/(8*(-1 + y^2/r^2)^(3/2)) + 21*i*r*y^2/(8*sqrt(-1 + y^2/r^2)), |y^2/r^2| > 1), (3*r^3/(8*sqrt(1 - y^2/r^2)) + 5*r^3*sqrt(1 - y^2/r^2)/8 - 5*r*y^2/(8*sqrt(1 - y^2/r^2)) - 3*r*y^2*sqrt(1 - y^2/r^2)/4 + y^4/(4*r*sqrt(1 - y^2/r^2)), True)), (y, 0, -r*sqrt(2)/2)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.