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