Sr Examen
Integral de x*sin(x/3)*cos(n*x) dx
Solución
Ha introducido
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pi / | | /x\ | x*sin|-|*cos(n*x) dx | \3/ | / -pi
$$\int\limits_{- \pi}^{\pi} x \sin{\left(\frac{x}{3} \right)} \cos{\left(n x \right)}\, dx$$
Integral((x*sin(x/3))*cos(n*x), (x, -pi, pi))
Respuesta
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/ ___ | 3*pi 9*\/ 3 | ---- + ------- for Or(n = -1/3, n = 1/3) | 4 8 | < ___ 2 ___ 2 ___ ___ 3 | 54*n*sin(pi*n) 3*pi*cos(pi*n) 9*\/ 3 *cos(pi*n) 27*pi*n *cos(pi*n) 81*\/ 3 *n *cos(pi*n) 9*pi*n*\/ 3 *sin(pi*n) 81*pi*\/ 3 *n *sin(pi*n) |- ----------------- - ----------------- + ----------------- + ------------------ + --------------------- - ---------------------- + ------------------------ otherwise | 2 4 2 4 2 4 2 4 2 4 2 4 2 4 | 1 - 18*n + 81*n 1 - 18*n + 81*n 1 - 18*n + 81*n 1 - 18*n + 81*n 1 - 18*n + 81*n 1 - 18*n + 81*n 1 - 18*n + 81*n \
$$\begin{cases} \frac{9 \sqrt{3}}{8} + \frac{3 \pi}{4} & \text{for}\: n = - \frac{1}{3} \vee n = \frac{1}{3} \\\frac{81 \sqrt{3} \pi n^{3} \sin{\left(\pi n \right)}}{81 n^{4} - 18 n^{2} + 1} + \frac{27 \pi n^{2} \cos{\left(\pi n \right)}}{81 n^{4} - 18 n^{2} + 1} + \frac{81 \sqrt{3} n^{2} \cos{\left(\pi n \right)}}{81 n^{4} - 18 n^{2} + 1} - \frac{54 n \sin{\left(\pi n \right)}}{81 n^{4} - 18 n^{2} + 1} - \frac{9 \sqrt{3} \pi n \sin{\left(\pi n \right)}}{81 n^{4} - 18 n^{2} + 1} - \frac{3 \pi \cos{\left(\pi n \right)}}{81 n^{4} - 18 n^{2} + 1} + \frac{9 \sqrt{3} \cos{\left(\pi n \right)}}{81 n^{4} - 18 n^{2} + 1} & \text{otherwise} \end{cases}$$
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/ ___ | 3*pi 9*\/ 3 | ---- + ------- for Or(n = -1/3, n = 1/3) | 4 8 | < ___ 2 ___ 2 ___ ___ 3 | 54*n*sin(pi*n) 3*pi*cos(pi*n) 9*\/ 3 *cos(pi*n) 27*pi*n *cos(pi*n) 81*\/ 3 *n *cos(pi*n) 9*pi*n*\/ 3 *sin(pi*n) 81*pi*\/ 3 *n *sin(pi*n) |- ----------------- - ----------------- + ----------------- + ------------------ + --------------------- - ---------------------- + ------------------------ otherwise | 2 4 2 4 2 4 2 4 2 4 2 4 2 4 | 1 - 18*n + 81*n 1 - 18*n + 81*n 1 - 18*n + 81*n 1 - 18*n + 81*n 1 - 18*n + 81*n 1 - 18*n + 81*n 1 - 18*n + 81*n \
$$\begin{cases} \frac{9 \sqrt{3}}{8} + \frac{3 \pi}{4} & \text{for}\: n = - \frac{1}{3} \vee n = \frac{1}{3} \\\frac{81 \sqrt{3} \pi n^{3} \sin{\left(\pi n \right)}}{81 n^{4} - 18 n^{2} + 1} + \frac{27 \pi n^{2} \cos{\left(\pi n \right)}}{81 n^{4} - 18 n^{2} + 1} + \frac{81 \sqrt{3} n^{2} \cos{\left(\pi n \right)}}{81 n^{4} - 18 n^{2} + 1} - \frac{54 n \sin{\left(\pi n \right)}}{81 n^{4} - 18 n^{2} + 1} - \frac{9 \sqrt{3} \pi n \sin{\left(\pi n \right)}}{81 n^{4} - 18 n^{2} + 1} - \frac{3 \pi \cos{\left(\pi n \right)}}{81 n^{4} - 18 n^{2} + 1} + \frac{9 \sqrt{3} \cos{\left(\pi n \right)}}{81 n^{4} - 18 n^{2} + 1} & \text{otherwise} \end{cases}$$
Piecewise((3*pi/4 + 9*sqrt(3)/8, (n = -1/3)∨(n = 1/3)), (-54*n*sin(pi*n)/(1 - 18*n^2 + 81*n^4) - 3*pi*cos(pi*n)/(1 - 18*n^2 + 81*n^4) + 9*sqrt(3)*cos(pi*n)/(1 - 18*n^2 + 81*n^4) + 27*pi*n^2*cos(pi*n)/(1 - 18*n^2 + 81*n^4) + 81*sqrt(3)*n^2*cos(pi*n)/(1 - 18*n^2 + 81*n^4) - 9*pi*n*sqrt(3)*sin(pi*n)/(1 - 18*n^2 + 81*n^4) + 81*pi*sqrt(3)*n^3*sin(pi*n)/(1 - 18*n^2 + 81*n^4), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.