Sr Examen
Integral de 2*x/(sqrt(4-x^2*cos(y)^2)) dx
Solución
Ha introducido
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2 / | | 2*x | ------------------- dx | ________________ | / 2 2 | \/ 4 - x *cos (y) | / 0
$$\int\limits_{0}^{2} \frac{2 x}{\sqrt{- x^{2} \cos^{2}{\left(y \right)} + 4}}\, dx$$
Integral((2*x)/sqrt(4 - x^2*cos(y)^2), (x, 0, 2))
Respuesta (Indefinida)
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// ________________ \
|| / 2 2 |
/ ||-2*\/ 4 - x *cos (y) 2 |
| ||---------------------- for cos (y) != 0|
| 2*x || 2 |
| ------------------- dx = C + |< cos (y) |
| ________________ || |
| / 2 2 || 2 |
| \/ 4 - x *cos (y) || x |
| || -- otherwise |
/ \\ 2 /
$$\int \frac{2 x}{\sqrt{- x^{2} \cos^{2}{\left(y \right)} + 4}}\, dx = C + \begin{cases} - \frac{2 \sqrt{- x^{2} \cos^{2}{\left(y \right)} + 4}}{\cos^{2}{\left(y \right)}} & \text{for}\: \cos^{2}{\left(y \right)} \neq 0 \\\frac{x^{2}}{2} & \text{otherwise} \end{cases}$$
Respuesta
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/ / pi 3*pi\ | 2 for Or|y = --, y = ----| | \ 2 2 / | | _______________ < / 2 | 4 2*\/ 4 - 4*cos (y) |------- - -------------------- otherwise | 2 2 |cos (y) cos (y) \
$$\begin{cases} 2 & \text{for}\: y = \frac{\pi}{2} \vee y = \frac{3 \pi}{2} \\- \frac{2 \sqrt{4 - 4 \cos^{2}{\left(y \right)}}}{\cos^{2}{\left(y \right)}} + \frac{4}{\cos^{2}{\left(y \right)}} & \text{otherwise} \end{cases}$$
=
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/ / pi 3*pi\ | 2 for Or|y = --, y = ----| | \ 2 2 / | | _______________ < / 2 | 4 2*\/ 4 - 4*cos (y) |------- - -------------------- otherwise | 2 2 |cos (y) cos (y) \
$$\begin{cases} 2 & \text{for}\: y = \frac{\pi}{2} \vee y = \frac{3 \pi}{2} \\- \frac{2 \sqrt{4 - 4 \cos^{2}{\left(y \right)}}}{\cos^{2}{\left(y \right)}} + \frac{4}{\cos^{2}{\left(y \right)}} & \text{otherwise} \end{cases}$$
Piecewise((2, (y = pi/2)∨(y = 3*pi/2)), (4/cos(y)^2 - 2*sqrt(4 - 4*cos(y)^2)/cos(y)^2, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.