Sr Examen
Integral de xcosyx^2sin^2yx dx
Solución
Ha introducido
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1 / | | 2 2 | x*cos (y*x)*sin (y*x) dx | / 0
$$\int\limits_{0}^{1} x \cos^{2}{\left(x y \right)} \sin^{2}{\left(x y \right)}\, dx$$
Integral((x*cos(y*x)^2)*sin(y*x)^2, (x, 0, 1))
Respuesta
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/ 4 4 4 4 2 2 3 3 |cos (y) sin (y) 1 cos (y) sin (y) cos (y)*sin (y) cos (y)*sin(y) sin (y)*cos(y) |------- + ------- + ----- - ------- - ------- + --------------- - -------------- + -------------- for And(y > -oo, y < oo, y != 0) < 16 16 2 2 2 8 8*y 8*y | 32*y 32*y 32*y | \ 0 otherwise
$$\begin{cases} \frac{\sin^{4}{\left(y \right)}}{16} + \frac{\sin^{2}{\left(y \right)} \cos^{2}{\left(y \right)}}{8} + \frac{\cos^{4}{\left(y \right)}}{16} + \frac{\sin^{3}{\left(y \right)} \cos{\left(y \right)}}{8 y} - \frac{\sin{\left(y \right)} \cos^{3}{\left(y \right)}}{8 y} - \frac{\sin^{4}{\left(y \right)}}{32 y^{2}} - \frac{\cos^{4}{\left(y \right)}}{32 y^{2}} + \frac{1}{32 y^{2}} & \text{for}\: y > -\infty \wedge y < \infty \wedge y \neq 0 \\0 & \text{otherwise} \end{cases}$$
=
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/ 4 4 4 4 2 2 3 3 |cos (y) sin (y) 1 cos (y) sin (y) cos (y)*sin (y) cos (y)*sin(y) sin (y)*cos(y) |------- + ------- + ----- - ------- - ------- + --------------- - -------------- + -------------- for And(y > -oo, y < oo, y != 0) < 16 16 2 2 2 8 8*y 8*y | 32*y 32*y 32*y | \ 0 otherwise
$$\begin{cases} \frac{\sin^{4}{\left(y \right)}}{16} + \frac{\sin^{2}{\left(y \right)} \cos^{2}{\left(y \right)}}{8} + \frac{\cos^{4}{\left(y \right)}}{16} + \frac{\sin^{3}{\left(y \right)} \cos{\left(y \right)}}{8 y} - \frac{\sin{\left(y \right)} \cos^{3}{\left(y \right)}}{8 y} - \frac{\sin^{4}{\left(y \right)}}{32 y^{2}} - \frac{\cos^{4}{\left(y \right)}}{32 y^{2}} + \frac{1}{32 y^{2}} & \text{for}\: y > -\infty \wedge y < \infty \wedge y \neq 0 \\0 & \text{otherwise} \end{cases}$$
Piecewise((cos(y)^4/16 + sin(y)^4/16 + 1/(32*y^2) - cos(y)^4/(32*y^2) - sin(y)^4/(32*y^2) + cos(y)^2*sin(y)^2/8 - cos(y)^3*sin(y)/(8*y) + sin(y)^3*cos(y)/(8*y), (y > -oo)∧(y < oo)∧(Ne(y, 0))), (0, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.