Sr Examen
Integral de (a^2-x^2)^1.5 dx
Solución
Ha introducido
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r / | | 3/2 | / 2 2\ | \a - x / dx | / 0
$$\int\limits_{0}^{r} \left(a^{2} - x^{2}\right)^{\frac{3}{2}}\, dx$$
Integral((a^2 - x^2)^(3/2), (x, 0, r))
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 /x\ \
|| I*a *acosh|-| 3 5 3 | 2| | // 2 /x\ \
|| \a/ 3*I*a*x I*x I*x*a |x | | || I*a *acosh|-| 3 | 2| |
||- ------------- - ----------------- + ------------------- + ----------------- for |--| > 1| || \a/ I*x I*a*x |x | |
|| 8 _________ _________ _________ | 2| | ||- ------------- + ------------------- - ----------------- for |--| > 1|
|| / 2 / 2 / 2 |a | | || 2 _________ _________ | 2| |
/ || / x / x / x | || / 2 / 2 |a | |
| || 8* / -1 + -- 4*a* / -1 + -- 8* / -1 + -- | || / x / x |
| 3/2 || / 2 / 2 / 2 | || 2*a* / -1 + -- 2* / -1 + -- |
| / 2 2\ || \/ a \/ a \/ a | 2 || / 2 / 2 |
| \a - x / dx = C - |< | + a *|< \/ a \/ a |
| || 4 /x\ | || |
/ || a *asin|-| 5 3 3 | || ________ |
|| \a/ x x*a 3*a*x | || / 2 |
|| ---------- - ------------------ - ---------------- + ---------------- otherwise | || / x |
|| 8 ________ ________ ________ | || 2 /x\ a*x* / 1 - -- |
|| / 2 / 2 / 2 | || a *asin|-| / 2 |
|| / x / x / x | || \a/ \/ a |
|| 4*a* / 1 - -- 8* / 1 - -- 8* / 1 - -- | || ---------- + ------------------ otherwise |
|| / 2 / 2 / 2 | \\ 2 2 /
\\ \/ a \/ a \/ a /
$$\int \left(a^{2} - x^{2}\right)^{\frac{3}{2}}\, dx = C + a^{2} \left(\begin{cases} - \frac{i a^{2} \operatorname{acosh}{\left(\frac{x}{a} \right)}}{2} - \frac{i a x}{2 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{i x^{3}}{2 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} & \text{for}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{a^{2} \operatorname{asin}{\left(\frac{x}{a} \right)}}{2} + \frac{a x \sqrt{1 - \frac{x^{2}}{a^{2}}}}{2} & \text{otherwise} \end{cases}\right) - \begin{cases} - \frac{i a^{4} \operatorname{acosh}{\left(\frac{x}{a} \right)}}{8} + \frac{i a^{3} x}{8 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} - \frac{3 i a x^{3}}{8 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{i x^{5}}{4 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} & \text{for}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{a^{4} \operatorname{asin}{\left(\frac{x}{a} \right)}}{8} - \frac{a^{3} x}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{3 a x^{3}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{x^{5}}{4 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}$$
Respuesta
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/ 0 | / | | | | / 3 4 4 6 2 2 | 2| | | | I*a 7*I*x 5*I*x I*x 5*I*a*x 21*I*a*x |x | | | |- --------------- - ---------------- - ------------------- + ----------------- + -------------- + ----------------- for |--| > 1 | | | _________ 3/2 _________ 3/2 3/2 _________ | 2| | | | / 2 / 2\ / 2 / 2\ / 2\ / 2 |a | | | | / x | x | / x 3 | x | | x | / x | | | / -1 + -- 8*a*|-1 + --| 4*a* / -1 + -- 4*a *|-1 + --| 8*|-1 + --| 8* / -1 + -- | | | / 2 | 2| / 2 | 2| | 2| / 2 | | | \/ a \ a / \/ a \ a / \ a / \/ a | | | | | | ________ ________ |- | < / 2 / 2 dx for r < 0 | | | 3 / x 2 / x | | | 5*a * / 1 - -- 3*a*x * / 1 - -- | | | 3 / 2 2 / 2 4 | | | 3*a \/ a 5*a*x \/ a x | | | ---------------- + ------------------- - ---------------- - --------------------- + ------------------ otherwise | | | ________ 8 ________ 4 ________ | | | / 2 / 2 / 2 | | | / x / x / x | | | 8* / 1 - -- 8* / 1 - -- 4*a* / 1 - -- | | | / 2 / 2 / 2 | | \ \/ a \/ a \/ a | | | / | r < | r | / | | | | / 3 4 4 6 2 2 | 2| | | | I*a 7*I*x 5*I*x I*x 5*I*a*x 21*I*a*x |x | | | |- --------------- - ---------------- - ------------------- + ----------------- + -------------- + ----------------- for |--| > 1 | | | _________ 3/2 _________ 3/2 3/2 _________ | 2| | | | / 2 / 2\ / 2 / 2\ / 2\ / 2 |a | | | | / x | x | / x 3 | x | | x | / x | | | / -1 + -- 8*a*|-1 + --| 4*a* / -1 + -- 4*a *|-1 + --| 8*|-1 + --| 8* / -1 + -- | | | / 2 | 2| / 2 | 2| | 2| / 2 | | | \/ a \ a / \/ a \ a / \ a / \/ a | | | | | | ________ ________ | | < / 2 / 2 dx otherwise | | | 3 / x 2 / x | | | 5*a * / 1 - -- 3*a*x * / 1 - -- | | | 3 / 2 2 / 2 4 | | | 3*a \/ a 5*a*x \/ a x | | | ---------------- + ------------------- - ---------------- - --------------------- + ------------------ otherwise | | | ________ 8 ________ 4 ________ | | | / 2 / 2 / 2 | | | / x / x / x | | | 8* / 1 - -- 8* / 1 - -- 4*a* / 1 - -- | | | / 2 / 2 / 2 | | \ \/ a \/ a \/ a | | |/ \0
$$\begin{cases} - \int\limits_{r}^{0} \begin{cases} - \frac{i a^{3}}{\sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{21 i a x^{2}}{8 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{5 i a x^{2}}{8 \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{5 i x^{4}}{4 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} - \frac{7 i x^{4}}{8 a \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} + \frac{i x^{6}}{4 a^{3} \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{5 a^{3} \sqrt{1 - \frac{x^{2}}{a^{2}}}}{8} + \frac{3 a^{3}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{3 a x^{2} \sqrt{1 - \frac{x^{2}}{a^{2}}}}{4} - \frac{5 a x^{2}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{x^{4}}{4 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\, dx & \text{for}\: r < 0 \\\int\limits_{0}^{r} \begin{cases} - \frac{i a^{3}}{\sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{21 i a x^{2}}{8 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{5 i a x^{2}}{8 \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{5 i x^{4}}{4 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} - \frac{7 i x^{4}}{8 a \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} + \frac{i x^{6}}{4 a^{3} \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{5 a^{3} \sqrt{1 - \frac{x^{2}}{a^{2}}}}{8} + \frac{3 a^{3}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{3 a x^{2} \sqrt{1 - \frac{x^{2}}{a^{2}}}}{4} - \frac{5 a x^{2}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{x^{4}}{4 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases}$$
=
=
/ 0 | / | | | | / 3 4 4 6 2 2 | 2| | | | I*a 7*I*x 5*I*x I*x 5*I*a*x 21*I*a*x |x | | | |- --------------- - ---------------- - ------------------- + ----------------- + -------------- + ----------------- for |--| > 1 | | | _________ 3/2 _________ 3/2 3/2 _________ | 2| | | | / 2 / 2\ / 2 / 2\ / 2\ / 2 |a | | | | / x | x | / x 3 | x | | x | / x | | | / -1 + -- 8*a*|-1 + --| 4*a* / -1 + -- 4*a *|-1 + --| 8*|-1 + --| 8* / -1 + -- | | | / 2 | 2| / 2 | 2| | 2| / 2 | | | \/ a \ a / \/ a \ a / \ a / \/ a | | | | | | ________ ________ |- | < / 2 / 2 dx for r < 0 | | | 3 / x 2 / x | | | 5*a * / 1 - -- 3*a*x * / 1 - -- | | | 3 / 2 2 / 2 4 | | | 3*a \/ a 5*a*x \/ a x | | | ---------------- + ------------------- - ---------------- - --------------------- + ------------------ otherwise | | | ________ 8 ________ 4 ________ | | | / 2 / 2 / 2 | | | / x / x / x | | | 8* / 1 - -- 8* / 1 - -- 4*a* / 1 - -- | | | / 2 / 2 / 2 | | \ \/ a \/ a \/ a | | | / | r < | r | / | | | | / 3 4 4 6 2 2 | 2| | | | I*a 7*I*x 5*I*x I*x 5*I*a*x 21*I*a*x |x | | | |- --------------- - ---------------- - ------------------- + ----------------- + -------------- + ----------------- for |--| > 1 | | | _________ 3/2 _________ 3/2 3/2 _________ | 2| | | | / 2 / 2\ / 2 / 2\ / 2\ / 2 |a | | | | / x | x | / x 3 | x | | x | / x | | | / -1 + -- 8*a*|-1 + --| 4*a* / -1 + -- 4*a *|-1 + --| 8*|-1 + --| 8* / -1 + -- | | | / 2 | 2| / 2 | 2| | 2| / 2 | | | \/ a \ a / \/ a \ a / \ a / \/ a | | | | | | ________ ________ | | < / 2 / 2 dx otherwise | | | 3 / x 2 / x | | | 5*a * / 1 - -- 3*a*x * / 1 - -- | | | 3 / 2 2 / 2 4 | | | 3*a \/ a 5*a*x \/ a x | | | ---------------- + ------------------- - ---------------- - --------------------- + ------------------ otherwise | | | ________ 8 ________ 4 ________ | | | / 2 / 2 / 2 | | | / x / x / x | | | 8* / 1 - -- 8* / 1 - -- 4*a* / 1 - -- | | | / 2 / 2 / 2 | | \ \/ a \/ a \/ a | | |/ \0
$$\begin{cases} - \int\limits_{r}^{0} \begin{cases} - \frac{i a^{3}}{\sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{21 i a x^{2}}{8 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{5 i a x^{2}}{8 \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{5 i x^{4}}{4 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} - \frac{7 i x^{4}}{8 a \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} + \frac{i x^{6}}{4 a^{3} \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{5 a^{3} \sqrt{1 - \frac{x^{2}}{a^{2}}}}{8} + \frac{3 a^{3}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{3 a x^{2} \sqrt{1 - \frac{x^{2}}{a^{2}}}}{4} - \frac{5 a x^{2}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{x^{4}}{4 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\, dx & \text{for}\: r < 0 \\\int\limits_{0}^{r} \begin{cases} - \frac{i a^{3}}{\sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{21 i a x^{2}}{8 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{5 i a x^{2}}{8 \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{5 i x^{4}}{4 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} - \frac{7 i x^{4}}{8 a \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} + \frac{i x^{6}}{4 a^{3} \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{5 a^{3} \sqrt{1 - \frac{x^{2}}{a^{2}}}}{8} + \frac{3 a^{3}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{3 a x^{2} \sqrt{1 - \frac{x^{2}}{a^{2}}}}{4} - \frac{5 a x^{2}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{x^{4}}{4 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases}$$
Piecewise((-Integral(Piecewise((-i*a^3/sqrt(-1 + x^2/a^2) - 7*i*x^4/(8*a*(-1 + x^2/a^2)^(3/2)) - 5*i*x^4/(4*a*sqrt(-1 + x^2/a^2)) + i*x^6/(4*a^3*(-1 + x^2/a^2)^(3/2)) + 5*i*a*x^2/(8*(-1 + x^2/a^2)^(3/2)) + 21*i*a*x^2/(8*sqrt(-1 + x^2/a^2)), |x^2/a^2| > 1), (3*a^3/(8*sqrt(1 - x^2/a^2)) + 5*a^3*sqrt(1 - x^2/a^2)/8 - 5*a*x^2/(8*sqrt(1 - x^2/a^2)) - 3*a*x^2*sqrt(1 - x^2/a^2)/4 + x^4/(4*a*sqrt(1 - x^2/a^2)), True)), (x, r, 0)), r < 0), (Integral(Piecewise((-i*a^3/sqrt(-1 + x^2/a^2) - 7*i*x^4/(8*a*(-1 + x^2/a^2)^(3/2)) - 5*i*x^4/(4*a*sqrt(-1 + x^2/a^2)) + i*x^6/(4*a^3*(-1 + x^2/a^2)^(3/2)) + 5*i*a*x^2/(8*(-1 + x^2/a^2)^(3/2)) + 21*i*a*x^2/(8*sqrt(-1 + x^2/a^2)), |x^2/a^2| > 1), (3*a^3/(8*sqrt(1 - x^2/a^2)) + 5*a^3*sqrt(1 - x^2/a^2)/8 - 5*a*x^2/(8*sqrt(1 - x^2/a^2)) - 3*a*x^2*sqrt(1 - x^2/a^2)/4 + x^4/(4*a*sqrt(1 - x^2/a^2)), True)), (x, 0, r)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.