1 / | | x | ---------- dx | 2 | x - x + 1 | / 0
Integral(x/(x^2 - x + 1), (x, 0, 1))
/ | | 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 | /
___ pi*\/ 3 -------- 9
=
___ pi*\/ 3 -------- 9
pi*sqrt(3)/9
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.