0 / | | 2 | --*x*(pi + x)*sin(n*x) dx | pi | / -pi
Integral((((2/pi)*x)*(pi + x))*sin(n*x), (x, -pi, 0))
// 0 for n = 0\ || | || //cos(n*x) x*sin(n*x) \ | / // 0 for n = 0\ \ || ||-------- + ---------- for n != 0| | | || | | || || 2 n | | | || //sin(n*x) \ | // 0 for n = 0\| // 0 for n = 0\ || || n | | | || ||-------- for n != 0| | || || 2 || | - 4*|<-|< | | + 2*pi*|- |<-|< n | | + x*|<-cos(n*x) || + 2*x *|<-cos(n*x) | || || 2 | | | || || | | ||---------- otherwise|| ||---------- otherwise| || || x | | | || \\ x otherwise / | \\ n /| \\ n / || || -- otherwise | | | ||------------------------- otherwise| | / || \\ 2 / | \ \\ n / / | ||-------------------------------------- otherwise| | 2 \\ n / | --*x*(pi + x)*sin(n*x) dx = C + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | pi pi | /
/ /2*cos(pi*n) pi*sin(pi*n)\ | 2*|----------- + ------------| | | 3 2 | | \ n n / 4 <- ------------------------------ + ----- for And(n > -oo, n < oo, n != 0) | pi 3 | pi*n | \ 0 otherwise
=
/ /2*cos(pi*n) pi*sin(pi*n)\ | 2*|----------- + ------------| | | 3 2 | | \ n n / 4 <- ------------------------------ + ----- for And(n > -oo, n < oo, n != 0) | pi 3 | pi*n | \ 0 otherwise
Piecewise((-2*(2*cos(pi*n)/n^3 + pi*sin(pi*n)/n^2)/pi + 4/(pi*n^3), (n > -oo)∧(n < oo)∧(Ne(n, 0))), (0, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.