Sr Examen
Integral de 4*(sin(x)-sin(3*x)/9+sin(5*x)/25)*sin(2*x*K)/pi dx
Solución
Ha introducido
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pi -- 2 / | | / sin(3*x) sin(5*x)\ | 4*|sin(x) - -------- + --------|*sin(2*x*k) | \ 9 25 / | ------------------------------------------- dx | pi | / -pi ---- 2
$$\int\limits_{- \frac{\pi}{2}}^{\frac{\pi}{2}} \frac{4 \left(\left(\sin{\left(x \right)} - \frac{\sin{\left(3 x \right)}}{9}\right) + \frac{\sin{\left(5 x \right)}}{25}\right) \sin{\left(k 2 x \right)}}{\pi}\, dx$$
Integral(((4*(sin(x) - sin(3*x)/9 + sin(5*x)/25))*sin((2*x)*k))/pi, (x, -pi/2, pi/2))
Respuesta
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/ -2/25 for k = -5/2 | | 2/9 for k = -3/2 | | -2 for k = -1/2 | | 2 for k = 1/2 | | -2/9 for k = 3/2 < | 2/25 for k = 5/2 | | 5 3 3 5 | 410648*k*cos(pi*k) 33152*k *cos(pi*k) 268480*k *cos(pi*k) 268480*k *cos(pi*k) 33152*k *cos(pi*k) 410648*k*cos(pi*k) |- ----------------------------------------- - ----------------------------------------- + ----------------------------------------- - ----------------------------------------- + ----------------------------------------- + ----------------------------------------- | 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 | -50625 - 126000*k + 14400*k + 233100*k -50625 - 126000*k + 14400*k + 233100*k -50625 - 126000*k + 14400*k + 233100*k -50625 - 126000*k + 14400*k + 233100*k -50625 - 126000*k + 14400*k + 233100*k -50625 - 126000*k + 14400*k + 233100*k |----------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------- otherwise \ pi pi
$$\begin{cases} - \frac{2}{25} & \text{for}\: k = - \frac{5}{2} \\\frac{2}{9} & \text{for}\: k = - \frac{3}{2} \\-2 & \text{for}\: k = - \frac{1}{2} \\2 & \text{for}\: k = \frac{1}{2} \\- \frac{2}{9} & \text{for}\: k = \frac{3}{2} \\\frac{2}{25} & \text{for}\: k = \frac{5}{2} \\\frac{- \frac{33152 k^{5} \cos{\left(\pi k \right)}}{14400 k^{6} - 126000 k^{4} + 233100 k^{2} - 50625} + \frac{268480 k^{3} \cos{\left(\pi k \right)}}{14400 k^{6} - 126000 k^{4} + 233100 k^{2} - 50625} - \frac{410648 k \cos{\left(\pi k \right)}}{14400 k^{6} - 126000 k^{4} + 233100 k^{2} - 50625}}{\pi} - \frac{\frac{33152 k^{5} \cos{\left(\pi k \right)}}{14400 k^{6} - 126000 k^{4} + 233100 k^{2} - 50625} - \frac{268480 k^{3} \cos{\left(\pi k \right)}}{14400 k^{6} - 126000 k^{4} + 233100 k^{2} - 50625} + \frac{410648 k \cos{\left(\pi k \right)}}{14400 k^{6} - 126000 k^{4} + 233100 k^{2} - 50625}}{\pi} & \text{otherwise} \end{cases}$$
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/ -2/25 for k = -5/2 | | 2/9 for k = -3/2 | | -2 for k = -1/2 | | 2 for k = 1/2 | | -2/9 for k = 3/2 < | 2/25 for k = 5/2 | | 5 3 3 5 | 410648*k*cos(pi*k) 33152*k *cos(pi*k) 268480*k *cos(pi*k) 268480*k *cos(pi*k) 33152*k *cos(pi*k) 410648*k*cos(pi*k) |- ----------------------------------------- - ----------------------------------------- + ----------------------------------------- - ----------------------------------------- + ----------------------------------------- + ----------------------------------------- | 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 | -50625 - 126000*k + 14400*k + 233100*k -50625 - 126000*k + 14400*k + 233100*k -50625 - 126000*k + 14400*k + 233100*k -50625 - 126000*k + 14400*k + 233100*k -50625 - 126000*k + 14400*k + 233100*k -50625 - 126000*k + 14400*k + 233100*k |----------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------- otherwise \ pi pi
$$\begin{cases} - \frac{2}{25} & \text{for}\: k = - \frac{5}{2} \\\frac{2}{9} & \text{for}\: k = - \frac{3}{2} \\-2 & \text{for}\: k = - \frac{1}{2} \\2 & \text{for}\: k = \frac{1}{2} \\- \frac{2}{9} & \text{for}\: k = \frac{3}{2} \\\frac{2}{25} & \text{for}\: k = \frac{5}{2} \\\frac{- \frac{33152 k^{5} \cos{\left(\pi k \right)}}{14400 k^{6} - 126000 k^{4} + 233100 k^{2} - 50625} + \frac{268480 k^{3} \cos{\left(\pi k \right)}}{14400 k^{6} - 126000 k^{4} + 233100 k^{2} - 50625} - \frac{410648 k \cos{\left(\pi k \right)}}{14400 k^{6} - 126000 k^{4} + 233100 k^{2} - 50625}}{\pi} - \frac{\frac{33152 k^{5} \cos{\left(\pi k \right)}}{14400 k^{6} - 126000 k^{4} + 233100 k^{2} - 50625} - \frac{268480 k^{3} \cos{\left(\pi k \right)}}{14400 k^{6} - 126000 k^{4} + 233100 k^{2} - 50625} + \frac{410648 k \cos{\left(\pi k \right)}}{14400 k^{6} - 126000 k^{4} + 233100 k^{2} - 50625}}{\pi} & \text{otherwise} \end{cases}$$
Piecewise((-2/25, k = -5/2), (2/9, k = -3/2), (-2, k = -1/2), (2, k = 1/2), (-2/9, k = 3/2), (2/25, k = 5/2), ((-410648*k*cos(pi*k)/(-50625 - 126000*k^4 + 14400*k^6 + 233100*k^2) - 33152*k^5*cos(pi*k)/(-50625 - 126000*k^4 + 14400*k^6 + 233100*k^2) + 268480*k^3*cos(pi*k)/(-50625 - 126000*k^4 + 14400*k^6 + 233100*k^2))/pi - (-268480*k^3*cos(pi*k)/(-50625 - 126000*k^4 + 14400*k^6 + 233100*k^2) + 33152*k^5*cos(pi*k)/(-50625 - 126000*k^4 + 14400*k^6 + 233100*k^2) + 410648*k*cos(pi*k)/(-50625 - 126000*k^4 + 14400*k^6 + 233100*k^2))/pi, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.