Sr Examen
Integral de x^2sqrt(a^2-x^2) dx
Solución
Ha introducido
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a / | | _________ | 2 / 2 2 | x *\/ a - x dx | / 0
$$\int\limits_{0}^{a} x^{2} \sqrt{a^{2} - x^{2}}\, dx$$
Integral(x^2*sqrt(a^2 - x^2), (x, 0, a))
Respuesta (Indefinida)
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// 4 /x\ \
|| I*a *acosh|-| 3 5 3 | 2| |
|| \a/ 3*I*a*x I*x I*x*a |x | |
||- ------------- - ----------------- + ------------------- + ----------------- for |--| > 1|
|| 8 _________ _________ _________ | 2| |
|| / 2 / 2 / 2 |a | |
/ || / x / x / x |
| || 8* / -1 + -- 4*a* / -1 + -- 8* / -1 + -- |
| _________ || / 2 / 2 / 2 |
| 2 / 2 2 || \/ a \/ a \/ a |
| x *\/ a - x dx = C + |< |
| || 4 /x\ |
/ || a *asin|-| 5 3 3 |
|| \a/ x x*a 3*a*x |
|| ---------- - ------------------ - ---------------- + ---------------- otherwise |
|| 8 ________ ________ ________ |
|| / 2 / 2 / 2 |
|| / x / x / x |
|| 4*a* / 1 - -- 8* / 1 - -- 8* / 1 - -- |
|| / 2 / 2 / 2 |
\\ \/ a \/ a \/ a /
$$\int x^{2} \sqrt{a^{2} - x^{2}}\, dx = C + \begin{cases} - \frac{i a^{4} \operatorname{acosh}{\left(\frac{x}{a} \right)}}{8} + \frac{i a^{3} x}{8 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} - \frac{3 i a x^{3}}{8 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{i x^{5}}{4 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} & \text{for}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{a^{4} \operatorname{asin}{\left(\frac{x}{a} \right)}}{8} - \frac{a^{3} x}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{3 a x^{3}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{x^{5}}{4 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}$$
Respuesta
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/ 0 | / | | | | / 2 6 2 4 4 | 2| | | | 9*I*a*x I*x I*a*x 3*I*x 5*I*x |x | | | |- ----------------- - ----------------- - -------------- + ---------------- + ------------------- for ---- > 1 | | | _________ 3/2 3/2 3/2 _________ 2 | | | / 2 / 2\ / 2\ / 2\ / 2 a | | | / x 3 | x | | x | | x | / x | | | 8* / -1 + -- 4*a *|-1 + --| 8*|-1 + --| 8*a*|-1 + --| 4*a* / -1 + -- | | | / 2 | 2| | 2| | 2| / 2 | | | \/ a \ a / \ a / \ a / \/ a |- | < dx for a < 0 | | | 4 6 2 4 2 | | | 5*x x a*x 3*x 9*a*x | | | - ------------------ - ---------------- - ------------- + --------------- + ---------------- otherwise | | | ________ 3/2 3/2 3/2 ________ | | | / 2 / 2\ / 2\ / 2\ / 2 | | | / x 3 | x | | x | | x | / x | | | 4*a* / 1 - -- 4*a *|1 - --| 8*|1 - --| 8*a*|1 - --| 8* / 1 - -- | | | / 2 | 2| | 2| | 2| / 2 | | \ \/ a \ a / \ a / \ a / \/ a | | | / | a < | a | / | | | | / 2 6 2 4 4 | 2| | | | 9*I*a*x I*x I*a*x 3*I*x 5*I*x |x | | | |- ----------------- - ----------------- - -------------- + ---------------- + ------------------- for ---- > 1 | | | _________ 3/2 3/2 3/2 _________ 2 | | | / 2 / 2\ / 2\ / 2\ / 2 a | | | / x 3 | x | | x | | x | / x | | | 8* / -1 + -- 4*a *|-1 + --| 8*|-1 + --| 8*a*|-1 + --| 4*a* / -1 + -- | | | / 2 | 2| | 2| | 2| / 2 | | | \/ a \ a / \ a / \ a / \/ a | | < dx otherwise | | | 4 6 2 4 2 | | | 5*x x a*x 3*x 9*a*x | | | - ------------------ - ---------------- - ------------- + --------------- + ---------------- otherwise | | | ________ 3/2 3/2 3/2 ________ | | | / 2 / 2\ / 2\ / 2\ / 2 | | | / x 3 | x | | x | | x | / x | | | 4*a* / 1 - -- 4*a *|1 - --| 8*|1 - --| 8*a*|1 - --| 8* / 1 - -- | | | / 2 | 2| | 2| | 2| / 2 | | \ \/ a \ a / \ a / \ a / \/ a | | |/ \0
$$\begin{cases} - \int\limits_{a}^{0} \begin{cases} - \frac{9 i a x^{2}}{8 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} - \frac{i a x^{2}}{8 \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{3 i x^{4}}{8 a \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{i x^{6}}{4 a^{3} \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{\left|{x^{2}}\right|}{a^{2}} > 1 \\\frac{9 a x^{2}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{a x^{2}}{8 \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{3 x^{4}}{8 a \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{x^{6}}{4 a^{3} \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx & \text{for}\: a < 0 \\\int\limits_{0}^{a} \begin{cases} - \frac{9 i a x^{2}}{8 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} - \frac{i a x^{2}}{8 \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{3 i x^{4}}{8 a \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{i x^{6}}{4 a^{3} \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{\left|{x^{2}}\right|}{a^{2}} > 1 \\\frac{9 a x^{2}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{a x^{2}}{8 \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{3 x^{4}}{8 a \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{x^{6}}{4 a^{3} \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases}$$
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/ 0 | / | | | | / 2 6 2 4 4 | 2| | | | 9*I*a*x I*x I*a*x 3*I*x 5*I*x |x | | | |- ----------------- - ----------------- - -------------- + ---------------- + ------------------- for ---- > 1 | | | _________ 3/2 3/2 3/2 _________ 2 | | | / 2 / 2\ / 2\ / 2\ / 2 a | | | / x 3 | x | | x | | x | / x | | | 8* / -1 + -- 4*a *|-1 + --| 8*|-1 + --| 8*a*|-1 + --| 4*a* / -1 + -- | | | / 2 | 2| | 2| | 2| / 2 | | | \/ a \ a / \ a / \ a / \/ a |- | < dx for a < 0 | | | 4 6 2 4 2 | | | 5*x x a*x 3*x 9*a*x | | | - ------------------ - ---------------- - ------------- + --------------- + ---------------- otherwise | | | ________ 3/2 3/2 3/2 ________ | | | / 2 / 2\ / 2\ / 2\ / 2 | | | / x 3 | x | | x | | x | / x | | | 4*a* / 1 - -- 4*a *|1 - --| 8*|1 - --| 8*a*|1 - --| 8* / 1 - -- | | | / 2 | 2| | 2| | 2| / 2 | | \ \/ a \ a / \ a / \ a / \/ a | | | / | a < | a | / | | | | / 2 6 2 4 4 | 2| | | | 9*I*a*x I*x I*a*x 3*I*x 5*I*x |x | | | |- ----------------- - ----------------- - -------------- + ---------------- + ------------------- for ---- > 1 | | | _________ 3/2 3/2 3/2 _________ 2 | | | / 2 / 2\ / 2\ / 2\ / 2 a | | | / x 3 | x | | x | | x | / x | | | 8* / -1 + -- 4*a *|-1 + --| 8*|-1 + --| 8*a*|-1 + --| 4*a* / -1 + -- | | | / 2 | 2| | 2| | 2| / 2 | | | \/ a \ a / \ a / \ a / \/ a | | < dx otherwise | | | 4 6 2 4 2 | | | 5*x x a*x 3*x 9*a*x | | | - ------------------ - ---------------- - ------------- + --------------- + ---------------- otherwise | | | ________ 3/2 3/2 3/2 ________ | | | / 2 / 2\ / 2\ / 2\ / 2 | | | / x 3 | x | | x | | x | / x | | | 4*a* / 1 - -- 4*a *|1 - --| 8*|1 - --| 8*a*|1 - --| 8* / 1 - -- | | | / 2 | 2| | 2| | 2| / 2 | | \ \/ a \ a / \ a / \ a / \/ a | | |/ \0
$$\begin{cases} - \int\limits_{a}^{0} \begin{cases} - \frac{9 i a x^{2}}{8 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} - \frac{i a x^{2}}{8 \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{3 i x^{4}}{8 a \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{i x^{6}}{4 a^{3} \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{\left|{x^{2}}\right|}{a^{2}} > 1 \\\frac{9 a x^{2}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{a x^{2}}{8 \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{3 x^{4}}{8 a \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{x^{6}}{4 a^{3} \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx & \text{for}\: a < 0 \\\int\limits_{0}^{a} \begin{cases} - \frac{9 i a x^{2}}{8 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} - \frac{i a x^{2}}{8 \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{3 i x^{4}}{8 a \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{i x^{6}}{4 a^{3} \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{\left|{x^{2}}\right|}{a^{2}} > 1 \\\frac{9 a x^{2}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{a x^{2}}{8 \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{3 x^{4}}{8 a \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{x^{6}}{4 a^{3} \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases}$$
Piecewise((-Integral(Piecewise((-9*i*a*x^2/(8*sqrt(-1 + x^2/a^2)) - i*x^6/(4*a^3*(-1 + x^2/a^2)^(3/2)) - i*a*x^2/(8*(-1 + x^2/a^2)^(3/2)) + 3*i*x^4/(8*a*(-1 + x^2/a^2)^(3/2)) + 5*i*x^4/(4*a*sqrt(-1 + x^2/a^2)), |x^2|/a^2 > 1), (-5*x^4/(4*a*sqrt(1 - x^2/a^2)) - x^6/(4*a^3*(1 - x^2/a^2)^(3/2)) - a*x^2/(8*(1 - x^2/a^2)^(3/2)) + 3*x^4/(8*a*(1 - x^2/a^2)^(3/2)) + 9*a*x^2/(8*sqrt(1 - x^2/a^2)), True)), (x, a, 0)), a < 0), (Integral(Piecewise((-9*i*a*x^2/(8*sqrt(-1 + x^2/a^2)) - i*x^6/(4*a^3*(-1 + x^2/a^2)^(3/2)) - i*a*x^2/(8*(-1 + x^2/a^2)^(3/2)) + 3*i*x^4/(8*a*(-1 + x^2/a^2)^(3/2)) + 5*i*x^4/(4*a*sqrt(-1 + x^2/a^2)), |x^2|/a^2 > 1), (-5*x^4/(4*a*sqrt(1 - x^2/a^2)) - x^6/(4*a^3*(1 - x^2/a^2)^(3/2)) - a*x^2/(8*(1 - x^2/a^2)^(3/2)) + 3*x^4/(8*a*(1 - x^2/a^2)^(3/2)) + 9*a*x^2/(8*sqrt(1 - x^2/a^2)), True)), (x, 0, a)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.