Sr Examen
Integral de (-x-1)*cos(pi/3*(1/2+k)x) dx
Solución
Ha introducido
[src]
3 / | | /pi \ | (-x - 1)*cos|--*(1/2 + k)*x| dx | \3 / | / 0
$$\int\limits_{0}^{3} \left(- x - 1\right) \cos{\left(x \frac{\pi}{3} \left(k + \frac{1}{2}\right) \right)}\, dx$$
Integral((-x - 1)*cos(((pi/3)*(1/2 + k))*x), (x, 0, 3))
Respuesta (Indefinida)
[src]
/// /pi*x pi*k*x\ \
// /pi*x pi*k*x\ /pi*x pi*k*x\ /pi*x pi*k*x\ \ |||6*sin|---- + ------| |
|| 36*cos|---- + ------| 6*pi*x*sin|---- + ------| 12*pi*k*x*sin|---- + ------| | ||| \ 6 3 / |
/ || \ 6 3 / \ 6 3 / \ 6 3 / | ||<-------------------- for k != -1/2 for pi + 2*pi*k = 0|
| ||------------------------ + ------------------------- + ---------------------------- for k != -1/2| ||| pi + 2*pi*k |
| /pi \ || 2 2 2 2 2 2 2 2 2 2 2 2 | ||| |
| (-x - 1)*cos|--*(1/2 + k)*x| dx = C - |
$$\int \left(- x - 1\right) \cos{\left(x \frac{\pi}{3} \left(k + \frac{1}{2}\right) \right)}\, dx = C - \begin{cases} \frac{12 \pi k x \sin{\left(\frac{\pi k x}{3} + \frac{\pi x}{6} \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} + \frac{6 \pi x \sin{\left(\frac{\pi k x}{3} + \frac{\pi x}{6} \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} + \frac{36 \cos{\left(\frac{\pi k x}{3} + \frac{\pi x}{6} \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} & \text{for}\: k \neq - \frac{1}{2} \\\frac{x^{2}}{2} & \text{otherwise} \end{cases} - \begin{cases} \begin{cases} \frac{6 \sin{\left(\frac{\pi k x}{3} + \frac{\pi x}{6} \right)}}{2 \pi k + \pi} & \text{for}\: k \neq - \frac{1}{2} \\x & \text{otherwise} \end{cases} & \text{for}\: 2 \pi k + \pi = 0 \\\frac{6 \sin{\left(\frac{\pi k x}{3} + \frac{\pi x}{6} \right)}}{2 \pi k + \pi} & \text{otherwise} \end{cases}$$
Respuesta
[src]
/ 36 6*cos(pi*k) 36*sin(pi*k) 18*pi*cos(pi*k) 36*pi*k*cos(pi*k) |------------------------ - ----------- + ------------------------ - ------------------------ - ------------------------ for And(k > -oo, k < oo, k != -1/2) | 2 2 2 2 pi + 2*pi*k 2 2 2 2 2 2 2 2 2 2 2 2 |pi + 4*k*pi + 4*pi *k pi + 4*k*pi + 4*pi *k pi + 4*k*pi + 4*pi *k pi + 4*k*pi + 4*pi *k | < -15/2 for k = -1/2 | | 9 6*cos(pi*k) | - - - ----------- otherwise | 2 pi + 2*pi*k \
$$\begin{cases} - \frac{36 \pi k \cos{\left(\pi k \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} + \frac{36 \sin{\left(\pi k \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} - \frac{18 \pi \cos{\left(\pi k \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} + \frac{36}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} - \frac{6 \cos{\left(\pi k \right)}}{2 \pi k + \pi} & \text{for}\: k > -\infty \wedge k < \infty \wedge k \neq - \frac{1}{2} \\- \frac{15}{2} & \text{for}\: k = - \frac{1}{2} \\- \frac{9}{2} - \frac{6 \cos{\left(\pi k \right)}}{2 \pi k + \pi} & \text{otherwise} \end{cases}$$
=
=
/ 36 6*cos(pi*k) 36*sin(pi*k) 18*pi*cos(pi*k) 36*pi*k*cos(pi*k) |------------------------ - ----------- + ------------------------ - ------------------------ - ------------------------ for And(k > -oo, k < oo, k != -1/2) | 2 2 2 2 pi + 2*pi*k 2 2 2 2 2 2 2 2 2 2 2 2 |pi + 4*k*pi + 4*pi *k pi + 4*k*pi + 4*pi *k pi + 4*k*pi + 4*pi *k pi + 4*k*pi + 4*pi *k | < -15/2 for k = -1/2 | | 9 6*cos(pi*k) | - - - ----------- otherwise | 2 pi + 2*pi*k \
$$\begin{cases} - \frac{36 \pi k \cos{\left(\pi k \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} + \frac{36 \sin{\left(\pi k \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} - \frac{18 \pi \cos{\left(\pi k \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} + \frac{36}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} - \frac{6 \cos{\left(\pi k \right)}}{2 \pi k + \pi} & \text{for}\: k > -\infty \wedge k < \infty \wedge k \neq - \frac{1}{2} \\- \frac{15}{2} & \text{for}\: k = - \frac{1}{2} \\- \frac{9}{2} - \frac{6 \cos{\left(\pi k \right)}}{2 \pi k + \pi} & \text{otherwise} \end{cases}$$
Piecewise((36/(pi^2 + 4*k*pi^2 + 4*pi^2*k^2) - 6*cos(pi*k)/(pi + 2*pi*k) + 36*sin(pi*k)/(pi^2 + 4*k*pi^2 + 4*pi^2*k^2) - 18*pi*cos(pi*k)/(pi^2 + 4*k*pi^2 + 4*pi^2*k^2) - 36*pi*k*cos(pi*k)/(pi^2 + 4*k*pi^2 + 4*pi^2*k^2), (k > -oo)∧(k < oo)∧(Ne(k, -1/2))), (-15/2, k = -1/2), (-9/2 - 6*cos(pi*k)/(pi + 2*pi*k), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.