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