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