1 / | | x | ------------ dx | 2 | x - 3*x + 3 | / 0
Integral(x/(x^2 - 3*x + 3), (x, 0, 1))
/ | | x | ------------ dx | 2 | x - 3*x + 3 | /
/ 2*x - 3 \ |------------| / 3 \ | 2 | |-----| x \x - 3*x + 3/ \2*3/4/ ------------ = -------------- + ------------------------- 2 2 2 x - 3*x + 3 / ___ \ |-2*\/ 3 ___| |--------*x + \/ 3 | + 1 \ 3 /
/ | | x | ------------ dx | 2 = | x - 3*x + 3 | /
/ | | 2*x - 3 | ------------ dx | 2 | x - 3*x + 3 / | | / | 1 ------------------ + 2* | ------------------------- dx 2 | 2 | / ___ \ | |-2*\/ 3 ___| | |--------*x + \/ 3 | + 1 | \ 3 / | /
/ | | 2*x - 3 | ------------ dx | 2 | x - 3*x + 3 | / ------------------ 2
2 u = x - 3*x
/ | | 1 | ----- du | 3 + u | / log(3 + u) ----------- = ---------- 2 2
/ | | 2*x - 3 | ------------ dx | 2 | x - 3*x + 3 | / 2 \ / log\3 + x - 3*x/ ------------------ = ----------------- 2 2
/ | | 1 2* | ------------------------- dx | 2 | / ___ \ | |-2*\/ 3 ___| | |--------*x + \/ 3 | + 1 | \ 3 / | /
___ ___ 2*x*\/ 3 v = \/ 3 - --------- 3
/ | | 1 2* | ------ dv = 2*atan(v) | 2 | 1 + v | /
/ | / ___\ | 1 ___ | ___ 2*x*\/ 3 | 2* | ------------------------- dx = \/ 3 *atan|- \/ 3 + ---------| | 2 \ 3 / | / ___ \ | |-2*\/ 3 ___| | |--------*x + \/ 3 | + 1 | \ 3 / | /
/ 2 \ / ___\ log\3 + x - 3*x/ ___ | ___ 2*x*\/ 3 | C + ----------------- + \/ 3 *atan|- \/ 3 + ---------| 2 \ 3 /
/ | / 2 \ / ___ \ | x log\3 + x - 3*x/ ___ |2*\/ 3 *(-3/2 + x)| | ------------ dx = C + ----------------- + \/ 3 *atan|------------------| | 2 2 \ 3 / | x - 3*x + 3 | /
___ 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.