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