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