1 / | | -x + 1 | ---------- dx | 2 | x + x + 1 | / 0
Integral((-x + 1)/(x^2 + x + 1), (x, 0, 1))
/ | | -x + 1 | ---------- dx | 2 | x + x + 1 | /
/ 2*x + 1 \ |----------| / 3 \ | 2 | |-----| -x + 1 \x + x + 1/ \2*3/4/ ---------- = - ------------ + ------------------------- 2 2 2 x + x + 1 / ___ ___\ |-2*\/ 3 \/ 3 | |--------*x - -----| + 1 \ 3 3 /
/ | | -x + 1 | ---------- dx | 2 = | x + x + 1 | /
/ | | 2*x + 1 | ---------- dx | 2 / | x + x + 1 | | | 1 / 2* | ------------------------- dx - ---------------- | 2 2 | / ___ ___\ | |-2*\/ 3 \/ 3 | | |--------*x - -----| + 1 | \ 3 3 / | /
/ | | 2*x + 1 - | ---------- dx | 2 | x + x + 1 | / ------------------ 2
2 u = x + x
/ | | 1 - | ----- du | 1 + u | / -log(1 + u) ------------- = ------------ 2 2
/ | | 2*x + 1 - | ---------- dx | 2 | x + x + 1 | / 2\ / -log\1 + x + x / ------------------ = ----------------- 2 2
/ | | 1 2* | ------------------------- dx | 2 | / ___ ___\ | |-2*\/ 3 \/ 3 | | |--------*x - -----| + 1 | \ 3 3 / | /
___ ___ \/ 3 2*x*\/ 3 v = - ----- - --------- 3 3
/ | | 1 2* | ------ dv = 2*atan(v) | 2 | 1 + v | /
/ | / ___ ___\ | 1 ___ |\/ 3 2*x*\/ 3 | 2* | ------------------------- dx = \/ 3 *atan|----- + ---------| | 2 \ 3 3 / | / ___ ___\ | |-2*\/ 3 \/ 3 | | |--------*x - -----| + 1 | \ 3 3 / | /
/ 2\ / ___ ___\ log\1 + x + x / ___ |\/ 3 2*x*\/ 3 | C - --------------- + \/ 3 *atan|----- + ---------| 2 \ 3 3 /
/ | / 2\ / ___ \ | -x + 1 log\1 + x + x / ___ |-2*\/ 3 *(-1/2 - x)| | ---------- dx = C - --------------- + \/ 3 *atan|-------------------| | 2 2 \ 3 / | x + x + 1 | /
___ log(3) pi*\/ 3 - ------ + -------- 2 6
=
___ log(3) pi*\/ 3 - ------ + -------- 2 6
-log(3)/2 + pi*sqrt(3)/6
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.