Sr Examen
Integral de sin(npix)(x-x²)/(npi) dx
Solución
Ha introducido
[src]
1 / | | / 2\ | sin(n*pi*x)*\x - x / | -------------------- dx | n*pi | / 0
$$\int\limits_{0}^{1} \frac{\left(- x^{2} + x\right) \sin{\left(x \pi n \right)}}{\pi n}\, dx$$
Integral((sin((n*pi)*x)*(x - x^2))/((n*pi)), (x, 0, 1))
Respuesta (Indefinida)
[src]
/ // 0 for n = 0\ \
| || | |
/ | // 0 for n = 0\ || //cos(pi*n*x) x*sin(pi*n*x) \ | |
| | || | || ||----------- + ------------- for n != 0| | |
| / 2\ | || //sin(pi*n*x) \ | || || 2 2 pi*n | | // 0 for n = 0\ // 0 for n = 0\|
| sin(n*pi*x)*\x - x / 1 | || ||----------- for pi*n != 0| | || || pi *n | | || | 2 || ||
| -------------------- dx = C + ----*|- |<-|< pi*n | | + 2*|<-|< | | + x*|<-cos(pi*n*x) | - x *|<-cos(pi*n*x) ||
| n*pi pi*n | || || | | || || 2 | | ||------------- otherwise| ||------------- otherwise||
| | || \\ x otherwise / | || || x | | \\ pi*n / \\ pi*n /|
/ | ||------------------------------- otherwise| || || -- otherwise | | |
| \\ pi*n / || \\ 2 / | |
| ||-------------------------------------------- otherwise| |
\ \\ pi*n / /
$$\int \frac{\left(- x^{2} + x\right) \sin{\left(x \pi n \right)}}{\pi n}\, dx = C + \frac{1}{\pi n} \left(- x^{2} \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\cos{\left(\pi n x \right)}}{\pi n} & \text{otherwise} \end{cases}\right) + x \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\cos{\left(\pi n x \right)}}{\pi n} & \text{otherwise} \end{cases}\right) - \begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\begin{cases} \frac{\sin{\left(\pi n x \right)}}{\pi n} & \text{for}\: \pi n \neq 0 \\x & \text{otherwise} \end{cases}}{\pi n} & \text{otherwise} \end{cases} + 2 \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\begin{cases} \frac{x \sin{\left(\pi n x \right)}}{\pi n} + \frac{\cos{\left(\pi n x \right)}}{\pi^{2} n^{2}} & \text{for}\: n \neq 0 \\\frac{x^{2}}{2} & \text{otherwise} \end{cases}}{\pi n} & \text{otherwise} \end{cases}\right)\right)$$
Respuesta
[src]
/ sin(pi*n) 2*cos(pi*n) | - --------- - ----------- | 2 2 3 3 | 2 pi *n pi *n <------ + ------------------------- for And(n > -oo, n < oo, n != 0) | 4 4 pi*n |pi *n | \ 0 otherwise
$$\begin{cases} \frac{- \frac{\sin{\left(\pi n \right)}}{\pi^{2} n^{2}} - \frac{2 \cos{\left(\pi n \right)}}{\pi^{3} n^{3}}}{\pi n} + \frac{2}{\pi^{4} n^{4}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases}$$
=
=
/ sin(pi*n) 2*cos(pi*n) | - --------- - ----------- | 2 2 3 3 | 2 pi *n pi *n <------ + ------------------------- for And(n > -oo, n < oo, n != 0) | 4 4 pi*n |pi *n | \ 0 otherwise
$$\begin{cases} \frac{- \frac{\sin{\left(\pi n \right)}}{\pi^{2} n^{2}} - \frac{2 \cos{\left(\pi n \right)}}{\pi^{3} n^{3}}}{\pi n} + \frac{2}{\pi^{4} n^{4}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases}$$
Piecewise((2/(pi^4*n^4) + (-sin(pi*n)/(pi^2*n^2) - 2*cos(pi*n)/(pi^3*n^3))/(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.