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