Sr Examen
Integral de sqrt(1-x^2)/x^6 dx
Solución
Ha introducido
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1 / | | ________ | / 2 | \/ 1 - x | ----------- dx | 6 | x | / ___ \/ 2
$$\int\limits_{\sqrt{2}}^{1} \frac{\sqrt{1 - x^{2}}}{x^{6}}\, dx$$
Integral(sqrt(1 - x^2)/x^6, (x, sqrt(2), 1))
Respuesta (Indefinida)
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/ | / | ________ | | / 2 | ___________________ | \/ 1 - x | \/ -(1 + x)*(-1 + x) | ----------- dx = C + | --------------------- dx | 6 | 6 | x | x | | / /
$$\int \frac{\sqrt{1 - x^{2}}}{x^{6}}\, dx = C + \int \frac{\sqrt{- \left(x - 1\right) \left(x + 1\right)}}{x^{6}}\, dx$$
Gráfica
Respuesta
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\/ 2
/
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| / _________ _________ _________
| | / 2 / 2 / 2
| | 2*I I*\/ -1 + x 2*I*\/ -1 + x I*\/ -1 + x I I 2
| |--------------- + -------------- - ---------------- - -------------- - ----------------- + ------------------ for x > 1
| | _________ 6 2 4 _________ _________
| | / 2 x 15*x 5*x 4 / 2 2 / 2
| |15*\/ -1 + x 5*x *\/ -1 + x 15*x *\/ -1 + x
- | < dx
| | ________ ________ ________
| | / 2 / 2 / 2
| | 2 \/ 1 - x 2*\/ 1 - x \/ 1 - x 1 1
| | - -------------- + ----------- - ------------- - ----------- - ----------------- + ---------------- otherwise
| | ________ 6 2 4 ________ ________
| | / 2 x 15*x 5*x 2 / 2 4 / 2
| \ 15*\/ 1 - x 15*x *\/ 1 - x 5*x *\/ 1 - x
|
/
1
$$- \int\limits_{1}^{\sqrt{2}} \begin{cases} \frac{2 i}{15 \sqrt{x^{2} - 1}} - \frac{2 i \sqrt{x^{2} - 1}}{15 x^{2}} + \frac{i}{15 x^{2} \sqrt{x^{2} - 1}} - \frac{i \sqrt{x^{2} - 1}}{5 x^{4}} - \frac{i}{5 x^{4} \sqrt{x^{2} - 1}} + \frac{i \sqrt{x^{2} - 1}}{x^{6}} & \text{for}\: x^{2} > 1 \\- \frac{2}{15 \sqrt{1 - x^{2}}} - \frac{2 \sqrt{1 - x^{2}}}{15 x^{2}} - \frac{1}{15 x^{2} \sqrt{1 - x^{2}}} - \frac{\sqrt{1 - x^{2}}}{5 x^{4}} + \frac{1}{5 x^{4} \sqrt{1 - x^{2}}} + \frac{\sqrt{1 - x^{2}}}{x^{6}} & \text{otherwise} \end{cases}\, dx$$
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\/ 2
/
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| / _________ _________ _________
| | / 2 / 2 / 2
| | 2*I I*\/ -1 + x 2*I*\/ -1 + x I*\/ -1 + x I I 2
| |--------------- + -------------- - ---------------- - -------------- - ----------------- + ------------------ for x > 1
| | _________ 6 2 4 _________ _________
| | / 2 x 15*x 5*x 4 / 2 2 / 2
| |15*\/ -1 + x 5*x *\/ -1 + x 15*x *\/ -1 + x
- | < dx
| | ________ ________ ________
| | / 2 / 2 / 2
| | 2 \/ 1 - x 2*\/ 1 - x \/ 1 - x 1 1
| | - -------------- + ----------- - ------------- - ----------- - ----------------- + ---------------- otherwise
| | ________ 6 2 4 ________ ________
| | / 2 x 15*x 5*x 2 / 2 4 / 2
| \ 15*\/ 1 - x 15*x *\/ 1 - x 5*x *\/ 1 - x
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/
1
$$- \int\limits_{1}^{\sqrt{2}} \begin{cases} \frac{2 i}{15 \sqrt{x^{2} - 1}} - \frac{2 i \sqrt{x^{2} - 1}}{15 x^{2}} + \frac{i}{15 x^{2} \sqrt{x^{2} - 1}} - \frac{i \sqrt{x^{2} - 1}}{5 x^{4}} - \frac{i}{5 x^{4} \sqrt{x^{2} - 1}} + \frac{i \sqrt{x^{2} - 1}}{x^{6}} & \text{for}\: x^{2} > 1 \\- \frac{2}{15 \sqrt{1 - x^{2}}} - \frac{2 \sqrt{1 - x^{2}}}{15 x^{2}} - \frac{1}{15 x^{2} \sqrt{1 - x^{2}}} - \frac{\sqrt{1 - x^{2}}}{5 x^{4}} + \frac{1}{5 x^{4} \sqrt{1 - x^{2}}} + \frac{\sqrt{1 - x^{2}}}{x^{6}} & \text{otherwise} \end{cases}\, dx$$
-Integral(Piecewise((2*i/(15*sqrt(-1 + x^2)) + i*sqrt(-1 + x^2)/x^6 - 2*i*sqrt(-1 + x^2)/(15*x^2) - i*sqrt(-1 + x^2)/(5*x^4) - i/(5*x^4*sqrt(-1 + x^2)) + i/(15*x^2*sqrt(-1 + x^2)), x^2 > 1), (-2/(15*sqrt(1 - x^2)) + sqrt(1 - x^2)/x^6 - 2*sqrt(1 - x^2)/(15*x^2) - sqrt(1 - x^2)/(5*x^4) - 1/(15*x^2*sqrt(1 - x^2)) + 1/(5*x^4*sqrt(1 - x^2)), True)), (x, 1, sqrt(2)))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.