Sr Examen
Integral de x^4(a^2-x^2)^1/2 dx
Solución
Ha introducido
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1 / | | _________ | 4 / 2 2 | x *\/ a - x dx | / 0
$$\int\limits_{0}^{1} x^{4} \sqrt{a^{2} - x^{2}}\, dx$$
Integral(x^4*sqrt(a^2 - x^2), (x, 0, 1))
Respuesta (Indefinida)
[src]
// 6 /x\ \
|| I*a *acosh|-| 5 3 3 7 5 | 2| |
|| \a/ 5*I*a*x I*a *x I*x I*x*a |x | |
||- ------------- - ------------------ - ------------------ + ------------------- + ------------------ for |--| > 1|
|| 16 _________ _________ _________ _________ | 2| |
|| / 2 / 2 / 2 / 2 |a | |
/ || / x / x / x / x |
| || 24* / -1 + -- 48* / -1 + -- 6*a* / -1 + -- 16* / -1 + -- |
| _________ || / 2 / 2 / 2 / 2 |
| 4 / 2 2 || \/ a \/ a \/ a \/ a |
| x *\/ a - x dx = C + |< |
| || 6 /x\ |
/ || a *asin|-| 7 5 3 3 5 |
|| \a/ x x*a a *x 5*a*x |
|| ---------- - ------------------ - ----------------- + ----------------- + ----------------- otherwise |
|| 16 ________ ________ ________ ________ |
|| / 2 / 2 / 2 / 2 |
|| / x / x / x / x |
|| 6*a* / 1 - -- 16* / 1 - -- 48* / 1 - -- 24* / 1 - -- |
|| / 2 / 2 / 2 / 2 |
\\ \/ a \/ a \/ a \/ a /
$$\int x^{4} \sqrt{a^{2} - x^{2}}\, dx = C + \begin{cases} - \frac{i a^{6} \operatorname{acosh}{\left(\frac{x}{a} \right)}}{16} + \frac{i a^{5} x}{16 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} - \frac{i a^{3} x^{3}}{48 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} - \frac{5 i a x^{5}}{24 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{i x^{7}}{6 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} & \text{for}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{a^{6} \operatorname{asin}{\left(\frac{x}{a} \right)}}{16} - \frac{a^{5} x}{16 \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{a^{3} x^{3}}{48 \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{5 a x^{5}}{24 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{x^{7}}{6 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}$$
Respuesta
[src]
1 / | | / 4 8 3 2 3 2 4 6 6 2 | | 25*I*a*x I*x I*a *x I*a *x I*a*x 5*I*x 7*I*x x | |- ------------------ - ----------------- - --------------- - ------------------ + --------------- + ----------------- + ------------------- for ---- > 1 | | _________ 3/2 3/2 _________ 3/2 3/2 _________ | 2| | | / 2 / 2\ / 2\ / 2 / 2\ / 2\ / 2 |a | | | / x 3 | x | | x | / x | x | | x | / x | | 24* / -1 + -- 6*a *|-1 + --| 16*|-1 + --| 16* / -1 + -- 48*|-1 + --| 24*a*|-1 + --| 6*a* / -1 + -- | | / 2 | 2| | 2| / 2 | 2| | 2| / 2 | | \/ a \ a / \ a / \/ a \ a / \ a / \/ a | < dx | | 6 8 3 2 3 2 4 6 4 | | 7*x x a *x a *x a*x 5*x 25*a*x | | - ------------------ - ---------------- - -------------- + ----------------- + -------------- + ---------------- + ----------------- otherwise | | ________ 3/2 3/2 ________ 3/2 3/2 ________ | | / 2 / 2\ / 2\ / 2 / 2\ / 2\ / 2 | | / x 3 | x | | x | / x | x | | x | / x | | 6*a* / 1 - -- 6*a *|1 - --| 16*|1 - --| 16* / 1 - -- 48*|1 - --| 24*a*|1 - --| 24* / 1 - -- | | / 2 | 2| | 2| / 2 | 2| | 2| / 2 | \ \/ a \ a / \ a / \/ a \ a / \ a / \/ a | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{i a^{3} x^{2}}{16 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} - \frac{i a^{3} x^{2}}{16 \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{25 i a x^{4}}{24 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{i a x^{4}}{48 \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} + \frac{7 i x^{6}}{6 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{5 i x^{6}}{24 a \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{i x^{8}}{6 a^{3} \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{x^{2}}{\left|{a^{2}}\right|} > 1 \\\frac{a^{3} x^{2}}{16 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{a^{3} x^{2}}{16 \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} + \frac{25 a x^{4}}{24 \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{a x^{4}}{48 \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{7 x^{6}}{6 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{5 x^{6}}{24 a \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{x^{8}}{6 a^{3} \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx$$
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1 / | | / 4 8 3 2 3 2 4 6 6 2 | | 25*I*a*x I*x I*a *x I*a *x I*a*x 5*I*x 7*I*x x | |- ------------------ - ----------------- - --------------- - ------------------ + --------------- + ----------------- + ------------------- for ---- > 1 | | _________ 3/2 3/2 _________ 3/2 3/2 _________ | 2| | | / 2 / 2\ / 2\ / 2 / 2\ / 2\ / 2 |a | | | / x 3 | x | | x | / x | x | | x | / x | | 24* / -1 + -- 6*a *|-1 + --| 16*|-1 + --| 16* / -1 + -- 48*|-1 + --| 24*a*|-1 + --| 6*a* / -1 + -- | | / 2 | 2| | 2| / 2 | 2| | 2| / 2 | | \/ a \ a / \ a / \/ a \ a / \ a / \/ a | < dx | | 6 8 3 2 3 2 4 6 4 | | 7*x x a *x a *x a*x 5*x 25*a*x | | - ------------------ - ---------------- - -------------- + ----------------- + -------------- + ---------------- + ----------------- otherwise | | ________ 3/2 3/2 ________ 3/2 3/2 ________ | | / 2 / 2\ / 2\ / 2 / 2\ / 2\ / 2 | | / x 3 | x | | x | / x | x | | x | / x | | 6*a* / 1 - -- 6*a *|1 - --| 16*|1 - --| 16* / 1 - -- 48*|1 - --| 24*a*|1 - --| 24* / 1 - -- | | / 2 | 2| | 2| / 2 | 2| | 2| / 2 | \ \/ a \ a / \ a / \/ a \ a / \ a / \/ a | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{i a^{3} x^{2}}{16 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} - \frac{i a^{3} x^{2}}{16 \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{25 i a x^{4}}{24 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{i a x^{4}}{48 \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} + \frac{7 i x^{6}}{6 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{5 i x^{6}}{24 a \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{i x^{8}}{6 a^{3} \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{x^{2}}{\left|{a^{2}}\right|} > 1 \\\frac{a^{3} x^{2}}{16 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{a^{3} x^{2}}{16 \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} + \frac{25 a x^{4}}{24 \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{a x^{4}}{48 \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{7 x^{6}}{6 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{5 x^{6}}{24 a \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{x^{8}}{6 a^{3} \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((-25*i*a*x^4/(24*sqrt(-1 + x^2/a^2)) - i*x^8/(6*a^3*(-1 + x^2/a^2)^(3/2)) - i*a^3*x^2/(16*(-1 + x^2/a^2)^(3/2)) - i*a^3*x^2/(16*sqrt(-1 + x^2/a^2)) + i*a*x^4/(48*(-1 + x^2/a^2)^(3/2)) + 5*i*x^6/(24*a*(-1 + x^2/a^2)^(3/2)) + 7*i*x^6/(6*a*sqrt(-1 + x^2/a^2)), x^2/|a^2| > 1), (-7*x^6/(6*a*sqrt(1 - x^2/a^2)) - x^8/(6*a^3*(1 - x^2/a^2)^(3/2)) - a^3*x^2/(16*(1 - x^2/a^2)^(3/2)) + a^3*x^2/(16*sqrt(1 - x^2/a^2)) + a*x^4/(48*(1 - x^2/a^2)^(3/2)) + 5*x^6/(24*a*(1 - x^2/a^2)^(3/2)) + 25*a*x^4/(24*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.