Sr Examen
Integral de (sin3x+sin3πx)sinπkx/5 dx
Solución
Ha introducido
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5 / | | (sin(3*x) + sin(3*pi*x))*sin(pi*k*x) | ------------------------------------ dx | 5 | / 0
$$\int\limits_{0}^{5} \frac{\left(\sin{\left(3 x \right)} + \sin{\left(3 \pi x \right)}\right) \sin{\left(x \pi k \right)}}{5}\, dx$$
Integral(((sin(3*x) + sin((3*pi)*x))*sin((pi*k)*x))/5, (x, 0, 5))
Respuesta
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/ 3 2 | 3*pi 3*pi 2*pi *sin(15) | - ------------- + ------------- - ----------------- for k = -3 | 3 3 / 3\ | -6*pi + 6*pi -6*pi + 6*pi 5*\-6*pi + 6*pi / | | 3 2 | 3*pi 3*pi 2*pi *sin(15) | - ------------- + ------------- + ----------------- for k = 3 | 3 3 / 3\ | -6*pi + 6*pi -6*pi + 6*pi 5*\-6*pi + 6*pi / | | 2 2 2 2 2 2 2 | 3*cos (15) 3*sin (15) 3*pi *cos (15) 3*pi *sin (15) 2*pi*sin(15) cos(15)*sin(15) pi *cos(15)*sin(15) -3 < ---------- + ---------- - -------------- - -------------- - -------------- - --------------- + ------------------- for k = --- | 2 2 2 2 / 2\ / 2\ / 2\ pi | -6 + 6*pi -6 + 6*pi -6 + 6*pi -6 + 6*pi 5*\-6 + 6*pi / 5*\-6 + 6*pi / 5*\-6 + 6*pi / | | 2 2 2 2 2 2 2 | 3*cos (15) 3*sin (15) 3*pi *cos (15) 3*pi *sin (15) cos(15)*sin(15) 2*pi*sin(15) pi *cos(15)*sin(15) 3 | - ---------- - ---------- + -------------- + -------------- + --------------- + -------------- - ------------------- for k = -- | 2 2 2 2 / 2\ / 2\ / 2\ pi | -6 + 6*pi -6 + 6*pi -6 + 6*pi -6 + 6*pi 5*\-6 + 6*pi / 5*\-6 + 6*pi / 5*\-6 + 6*pi / | | 2 2 2 3 2 2 | 27*sin(5*pi*k) 27*pi*cos(15)*sin(5*pi*k) 3*pi *k *sin(5*pi*k) pi *k *cos(5*pi*k)*sin(15) 3*pi*k *cos(15)*sin(5*pi*k) 9*k*pi *cos(5*pi*k)*sin(15) |--------------------------------------- - --------------------------------------- - --------------------------------------- - --------------------------------------- + --------------------------------------- + --------------------------------------- otherwise | / 3 4 2 3 2\ / 3 4 2 3 2\ / 3 4 2 3 2\ / 3 4 2 3 2\ / 3 4 2 3 2\ / 3 4 2 3 2\ \5*\81*pi + pi *k - 9*pi*k - 9*pi *k / 5*\81*pi + pi *k - 9*pi*k - 9*pi *k / 5*\81*pi + pi *k - 9*pi*k - 9*pi *k / 5*\81*pi + pi *k - 9*pi*k - 9*pi *k / 5*\81*pi + pi *k - 9*pi*k - 9*pi *k / 5*\81*pi + pi *k - 9*pi*k - 9*pi *k /
$$\begin{cases} - \frac{3 \pi^{3}}{- 6 \pi + 6 \pi^{3}} - \frac{2 \pi^{2} \sin{\left(15 \right)}}{5 \left(- 6 \pi + 6 \pi^{3}\right)} + \frac{3 \pi}{- 6 \pi + 6 \pi^{3}} & \text{for}\: k = -3 \\- \frac{3 \pi}{- 6 \pi + 6 \pi^{3}} + \frac{2 \pi^{2} \sin{\left(15 \right)}}{5 \left(- 6 \pi + 6 \pi^{3}\right)} + \frac{3 \pi^{3}}{- 6 \pi + 6 \pi^{3}} & \text{for}\: k = 3 \\- \frac{3 \pi^{2} \cos^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} - \frac{3 \pi^{2} \sin^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} + \frac{\pi^{2} \sin{\left(15 \right)} \cos{\left(15 \right)}}{5 \left(-6 + 6 \pi^{2}\right)} - \frac{2 \pi \sin{\left(15 \right)}}{5 \left(-6 + 6 \pi^{2}\right)} - \frac{\sin{\left(15 \right)} \cos{\left(15 \right)}}{5 \left(-6 + 6 \pi^{2}\right)} + \frac{3 \sin^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} + \frac{3 \cos^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} & \text{for}\: k = - \frac{3}{\pi} \\- \frac{3 \cos^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} - \frac{3 \sin^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} + \frac{\sin{\left(15 \right)} \cos{\left(15 \right)}}{5 \left(-6 + 6 \pi^{2}\right)} + \frac{2 \pi \sin{\left(15 \right)}}{5 \left(-6 + 6 \pi^{2}\right)} - \frac{\pi^{2} \sin{\left(15 \right)} \cos{\left(15 \right)}}{5 \left(-6 + 6 \pi^{2}\right)} + \frac{3 \pi^{2} \sin^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} + \frac{3 \pi^{2} \cos^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} & \text{for}\: k = \frac{3}{\pi} \\- \frac{\pi^{2} k^{3} \sin{\left(15 \right)} \cos{\left(5 \pi k \right)}}{5 \left(\pi^{3} k^{4} - 9 \pi^{3} k^{2} - 9 \pi k^{2} + 81 \pi\right)} - \frac{3 \pi^{2} k^{2} \sin{\left(5 \pi k \right)}}{5 \left(\pi^{3} k^{4} - 9 \pi^{3} k^{2} - 9 \pi k^{2} + 81 \pi\right)} + \frac{3 \pi k^{2} \sin{\left(5 \pi k \right)} \cos{\left(15 \right)}}{5 \left(\pi^{3} k^{4} - 9 \pi^{3} k^{2} - 9 \pi k^{2} + 81 \pi\right)} + \frac{9 \pi^{2} k \sin{\left(15 \right)} \cos{\left(5 \pi k \right)}}{5 \left(\pi^{3} k^{4} - 9 \pi^{3} k^{2} - 9 \pi k^{2} + 81 \pi\right)} + \frac{27 \sin{\left(5 \pi k \right)}}{5 \left(\pi^{3} k^{4} - 9 \pi^{3} k^{2} - 9 \pi k^{2} + 81 \pi\right)} - \frac{27 \pi \sin{\left(5 \pi k \right)} \cos{\left(15 \right)}}{5 \left(\pi^{3} k^{4} - 9 \pi^{3} k^{2} - 9 \pi k^{2} + 81 \pi\right)} & \text{otherwise} \end{cases}$$
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/ 3 2 | 3*pi 3*pi 2*pi *sin(15) | - ------------- + ------------- - ----------------- for k = -3 | 3 3 / 3\ | -6*pi + 6*pi -6*pi + 6*pi 5*\-6*pi + 6*pi / | | 3 2 | 3*pi 3*pi 2*pi *sin(15) | - ------------- + ------------- + ----------------- for k = 3 | 3 3 / 3\ | -6*pi + 6*pi -6*pi + 6*pi 5*\-6*pi + 6*pi / | | 2 2 2 2 2 2 2 | 3*cos (15) 3*sin (15) 3*pi *cos (15) 3*pi *sin (15) 2*pi*sin(15) cos(15)*sin(15) pi *cos(15)*sin(15) -3 < ---------- + ---------- - -------------- - -------------- - -------------- - --------------- + ------------------- for k = --- | 2 2 2 2 / 2\ / 2\ / 2\ pi | -6 + 6*pi -6 + 6*pi -6 + 6*pi -6 + 6*pi 5*\-6 + 6*pi / 5*\-6 + 6*pi / 5*\-6 + 6*pi / | | 2 2 2 2 2 2 2 | 3*cos (15) 3*sin (15) 3*pi *cos (15) 3*pi *sin (15) cos(15)*sin(15) 2*pi*sin(15) pi *cos(15)*sin(15) 3 | - ---------- - ---------- + -------------- + -------------- + --------------- + -------------- - ------------------- for k = -- | 2 2 2 2 / 2\ / 2\ / 2\ pi | -6 + 6*pi -6 + 6*pi -6 + 6*pi -6 + 6*pi 5*\-6 + 6*pi / 5*\-6 + 6*pi / 5*\-6 + 6*pi / | | 2 2 2 3 2 2 | 27*sin(5*pi*k) 27*pi*cos(15)*sin(5*pi*k) 3*pi *k *sin(5*pi*k) pi *k *cos(5*pi*k)*sin(15) 3*pi*k *cos(15)*sin(5*pi*k) 9*k*pi *cos(5*pi*k)*sin(15) |--------------------------------------- - --------------------------------------- - --------------------------------------- - --------------------------------------- + --------------------------------------- + --------------------------------------- otherwise | / 3 4 2 3 2\ / 3 4 2 3 2\ / 3 4 2 3 2\ / 3 4 2 3 2\ / 3 4 2 3 2\ / 3 4 2 3 2\ \5*\81*pi + pi *k - 9*pi*k - 9*pi *k / 5*\81*pi + pi *k - 9*pi*k - 9*pi *k / 5*\81*pi + pi *k - 9*pi*k - 9*pi *k / 5*\81*pi + pi *k - 9*pi*k - 9*pi *k / 5*\81*pi + pi *k - 9*pi*k - 9*pi *k / 5*\81*pi + pi *k - 9*pi*k - 9*pi *k /
$$\begin{cases} - \frac{3 \pi^{3}}{- 6 \pi + 6 \pi^{3}} - \frac{2 \pi^{2} \sin{\left(15 \right)}}{5 \left(- 6 \pi + 6 \pi^{3}\right)} + \frac{3 \pi}{- 6 \pi + 6 \pi^{3}} & \text{for}\: k = -3 \\- \frac{3 \pi}{- 6 \pi + 6 \pi^{3}} + \frac{2 \pi^{2} \sin{\left(15 \right)}}{5 \left(- 6 \pi + 6 \pi^{3}\right)} + \frac{3 \pi^{3}}{- 6 \pi + 6 \pi^{3}} & \text{for}\: k = 3 \\- \frac{3 \pi^{2} \cos^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} - \frac{3 \pi^{2} \sin^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} + \frac{\pi^{2} \sin{\left(15 \right)} \cos{\left(15 \right)}}{5 \left(-6 + 6 \pi^{2}\right)} - \frac{2 \pi \sin{\left(15 \right)}}{5 \left(-6 + 6 \pi^{2}\right)} - \frac{\sin{\left(15 \right)} \cos{\left(15 \right)}}{5 \left(-6 + 6 \pi^{2}\right)} + \frac{3 \sin^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} + \frac{3 \cos^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} & \text{for}\: k = - \frac{3}{\pi} \\- \frac{3 \cos^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} - \frac{3 \sin^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} + \frac{\sin{\left(15 \right)} \cos{\left(15 \right)}}{5 \left(-6 + 6 \pi^{2}\right)} + \frac{2 \pi \sin{\left(15 \right)}}{5 \left(-6 + 6 \pi^{2}\right)} - \frac{\pi^{2} \sin{\left(15 \right)} \cos{\left(15 \right)}}{5 \left(-6 + 6 \pi^{2}\right)} + \frac{3 \pi^{2} \sin^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} + \frac{3 \pi^{2} \cos^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} & \text{for}\: k = \frac{3}{\pi} \\- \frac{\pi^{2} k^{3} \sin{\left(15 \right)} \cos{\left(5 \pi k \right)}}{5 \left(\pi^{3} k^{4} - 9 \pi^{3} k^{2} - 9 \pi k^{2} + 81 \pi\right)} - \frac{3 \pi^{2} k^{2} \sin{\left(5 \pi k \right)}}{5 \left(\pi^{3} k^{4} - 9 \pi^{3} k^{2} - 9 \pi k^{2} + 81 \pi\right)} + \frac{3 \pi k^{2} \sin{\left(5 \pi k \right)} \cos{\left(15 \right)}}{5 \left(\pi^{3} k^{4} - 9 \pi^{3} k^{2} - 9 \pi k^{2} + 81 \pi\right)} + \frac{9 \pi^{2} k \sin{\left(15 \right)} \cos{\left(5 \pi k \right)}}{5 \left(\pi^{3} k^{4} - 9 \pi^{3} k^{2} - 9 \pi k^{2} + 81 \pi\right)} + \frac{27 \sin{\left(5 \pi k \right)}}{5 \left(\pi^{3} k^{4} - 9 \pi^{3} k^{2} - 9 \pi k^{2} + 81 \pi\right)} - \frac{27 \pi \sin{\left(5 \pi k \right)} \cos{\left(15 \right)}}{5 \left(\pi^{3} k^{4} - 9 \pi^{3} k^{2} - 9 \pi k^{2} + 81 \pi\right)} & \text{otherwise} \end{cases}$$
Piecewise((-3*pi^3/(-6*pi + 6*pi^3) + 3*pi/(-6*pi + 6*pi^3) - 2*pi^2*sin(15)/(5*(-6*pi + 6*pi^3)), k = -3), (-3*pi/(-6*pi + 6*pi^3) + 3*pi^3/(-6*pi + 6*pi^3) + 2*pi^2*sin(15)/(5*(-6*pi + 6*pi^3)), k = 3), (3*cos(15)^2/(-6 + 6*pi^2) + 3*sin(15)^2/(-6 + 6*pi^2) - 3*pi^2*cos(15)^2/(-6 + 6*pi^2) - 3*pi^2*sin(15)^2/(-6 + 6*pi^2) - 2*pi*sin(15)/(5*(-6 + 6*pi^2)) - cos(15)*sin(15)/(5*(-6 + 6*pi^2)) + pi^2*cos(15)*sin(15)/(5*(-6 + 6*pi^2)), k = -3/pi), (-3*cos(15)^2/(-6 + 6*pi^2) - 3*sin(15)^2/(-6 + 6*pi^2) + 3*pi^2*cos(15)^2/(-6 + 6*pi^2) + 3*pi^2*sin(15)^2/(-6 + 6*pi^2) + cos(15)*sin(15)/(5*(-6 + 6*pi^2)) + 2*pi*sin(15)/(5*(-6 + 6*pi^2)) - pi^2*cos(15)*sin(15)/(5*(-6 + 6*pi^2)), k = 3/pi), (27*sin(5*pi*k)/(5*(81*pi + pi^3*k^4 - 9*pi*k^2 - 9*pi^3*k^2)) - 27*pi*cos(15)*sin(5*pi*k)/(5*(81*pi + pi^3*k^4 - 9*pi*k^2 - 9*pi^3*k^2)) - 3*pi^2*k^2*sin(5*pi*k)/(5*(81*pi + pi^3*k^4 - 9*pi*k^2 - 9*pi^3*k^2)) - pi^2*k^3*cos(5*pi*k)*sin(15)/(5*(81*pi + pi^3*k^4 - 9*pi*k^2 - 9*pi^3*k^2)) + 3*pi*k^2*cos(15)*sin(5*pi*k)/(5*(81*pi + pi^3*k^4 - 9*pi*k^2 - 9*pi^3*k^2)) + 9*k*pi^2*cos(5*pi*k)*sin(15)/(5*(81*pi + pi^3*k^4 - 9*pi*k^2 - 9*pi^3*k^2)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.