Sr Examen
Integral de (0.5*sin(2*x)-x/pi)*sin(k*x) dx
Solución
Ha introducido
[src]
pi / | | /sin(2*x) x \ | |-------- - --|*sin(k*x) dx | \ 2 pi/ | / 0
$$\int\limits_{0}^{\pi} \left(- \frac{x}{\pi} + \frac{\sin{\left(2 x \right)}}{2}\right) \sin{\left(k x \right)}\, dx$$
Integral((sin(2*x)/2 - x/pi)*sin(k*x), (x, 0, pi))
Respuesta (Indefinida)
[src]
// 2 2 \
|| x*cos (x)*cos(2*x) x*sin (x)*cos(2*x) cos(x)*cos(2*x)*sin(x) x*cos(x)*sin(x)*sin(2*x) |
||- ------------------ + ------------------ + ---------------------- - ------------------------ for k = -2|
// 0 for k = 0\ || 4 4 4 2 |
|| | || |
|| //sin(k*x) \ | || 2 2 | // 0 for k = 0\
|| ||-------- for k != 0| | || x*sin (x)*cos(2*x) cos(x)*cos(2*x)*sin(x) x*cos (x)*cos(2*x) x*cos(x)*sin(x)*sin(2*x) | || |
- 2*|<-|< k | | - 2*pi*|<- ------------------ - ---------------------- + ------------------ + ------------------------ for k = 2 | + 2*x*|<-cos(k*x) |
|| || | | || 4 4 4 2 | ||---------- otherwise|
|| \\ x otherwise / | || | \\ k /
||------------------------- otherwise| || 2 2 |
\\ k / || cos (x)*sin(k*x) sin (x)*sin(k*x) k*cos(x)*cos(k*x)*sin(x) |
/ || ---------------- - ---------------- - ------------------------ otherwise |
| || 2 2 2 |
| /sin(2*x) x \ \\ -4 + k -4 + k -4 + k /
| |-------- - --|*sin(k*x) dx = C - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
| \ 2 pi/ 2*pi
|
/
$$\int \left(- \frac{x}{\pi} + \frac{\sin{\left(2 x \right)}}{2}\right) \sin{\left(k x \right)}\, dx = C - \frac{2 x \left(\begin{cases} 0 & \text{for}\: k = 0 \\- \frac{\cos{\left(k x \right)}}{k} & \text{otherwise} \end{cases}\right) - 2 \left(\begin{cases} 0 & \text{for}\: k = 0 \\- \frac{\begin{cases} \frac{\sin{\left(k x \right)}}{k} & \text{for}\: k \neq 0 \\x & \text{otherwise} \end{cases}}{k} & \text{otherwise} \end{cases}\right) - 2 \pi \left(\begin{cases} \frac{x \sin^{2}{\left(x \right)} \cos{\left(2 x \right)}}{4} - \frac{x \sin{\left(x \right)} \sin{\left(2 x \right)} \cos{\left(x \right)}}{2} - \frac{x \cos^{2}{\left(x \right)} \cos{\left(2 x \right)}}{4} + \frac{\sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(2 x \right)}}{4} & \text{for}\: k = -2 \\- \frac{x \sin^{2}{\left(x \right)} \cos{\left(2 x \right)}}{4} + \frac{x \sin{\left(x \right)} \sin{\left(2 x \right)} \cos{\left(x \right)}}{2} + \frac{x \cos^{2}{\left(x \right)} \cos{\left(2 x \right)}}{4} - \frac{\sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(2 x \right)}}{4} & \text{for}\: k = 2 \\- \frac{k \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(k x \right)}}{k^{2} - 4} - \frac{\sin^{2}{\left(x \right)} \sin{\left(k x \right)}}{k^{2} - 4} + \frac{\sin{\left(k x \right)} \cos^{2}{\left(x \right)}}{k^{2} - 4} & \text{otherwise} \end{cases}\right)}{2 \pi}$$
Respuesta
[src]
/ 1 pi | - - - -- for k = -2 | 2 4 | | 0 for k = 0 | | 1 pi < - + -- for k = 2 | 2 4 | | 2 2 3 | 8*sin(pi*k) 2*k *sin(pi*k) 8*pi*k*cos(pi*k) 2*pi*k *sin(pi*k) 2*pi*k *cos(pi*k) |------------------- - ------------------- - ------------------- + ------------------- + ------------------- otherwise | 2 4 2 4 2 4 2 4 2 4 \- 8*pi*k + 2*pi*k - 8*pi*k + 2*pi*k - 8*pi*k + 2*pi*k - 8*pi*k + 2*pi*k - 8*pi*k + 2*pi*k
$$\begin{cases} - \frac{\pi}{4} - \frac{1}{2} & \text{for}\: k = -2 \\0 & \text{for}\: k = 0 \\\frac{1}{2} + \frac{\pi}{4} & \text{for}\: k = 2 \\\frac{2 \pi k^{3} \cos{\left(\pi k \right)}}{2 \pi k^{4} - 8 \pi k^{2}} - \frac{2 k^{2} \sin{\left(\pi k \right)}}{2 \pi k^{4} - 8 \pi k^{2}} + \frac{2 \pi k^{2} \sin{\left(\pi k \right)}}{2 \pi k^{4} - 8 \pi k^{2}} - \frac{8 \pi k \cos{\left(\pi k \right)}}{2 \pi k^{4} - 8 \pi k^{2}} + \frac{8 \sin{\left(\pi k \right)}}{2 \pi k^{4} - 8 \pi k^{2}} & \text{otherwise} \end{cases}$$
=
=
/ 1 pi | - - - -- for k = -2 | 2 4 | | 0 for k = 0 | | 1 pi < - + -- for k = 2 | 2 4 | | 2 2 3 | 8*sin(pi*k) 2*k *sin(pi*k) 8*pi*k*cos(pi*k) 2*pi*k *sin(pi*k) 2*pi*k *cos(pi*k) |------------------- - ------------------- - ------------------- + ------------------- + ------------------- otherwise | 2 4 2 4 2 4 2 4 2 4 \- 8*pi*k + 2*pi*k - 8*pi*k + 2*pi*k - 8*pi*k + 2*pi*k - 8*pi*k + 2*pi*k - 8*pi*k + 2*pi*k
$$\begin{cases} - \frac{\pi}{4} - \frac{1}{2} & \text{for}\: k = -2 \\0 & \text{for}\: k = 0 \\\frac{1}{2} + \frac{\pi}{4} & \text{for}\: k = 2 \\\frac{2 \pi k^{3} \cos{\left(\pi k \right)}}{2 \pi k^{4} - 8 \pi k^{2}} - \frac{2 k^{2} \sin{\left(\pi k \right)}}{2 \pi k^{4} - 8 \pi k^{2}} + \frac{2 \pi k^{2} \sin{\left(\pi k \right)}}{2 \pi k^{4} - 8 \pi k^{2}} - \frac{8 \pi k \cos{\left(\pi k \right)}}{2 \pi k^{4} - 8 \pi k^{2}} + \frac{8 \sin{\left(\pi k \right)}}{2 \pi k^{4} - 8 \pi k^{2}} & \text{otherwise} \end{cases}$$
Piecewise((-1/2 - pi/4, k = -2), (0, k = 0), (1/2 + pi/4, k = 2), (8*sin(pi*k)/(-8*pi*k^2 + 2*pi*k^4) - 2*k^2*sin(pi*k)/(-8*pi*k^2 + 2*pi*k^4) - 8*pi*k*cos(pi*k)/(-8*pi*k^2 + 2*pi*k^4) + 2*pi*k^2*sin(pi*k)/(-8*pi*k^2 + 2*pi*k^4) + 2*pi*k^3*cos(pi*k)/(-8*pi*k^2 + 2*pi*k^4), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.