1 / | | 3*x - 2 | ------------ dx | 2 | x - 4*x + 5 | / 0
Integral((3*x - 2)/(x^2 - 4*x + 5), (x, 0, 1))
/ | | 3*x - 2 | ------------ dx | 2 | x - 4*x + 5 | /
2*x - 4 3*------------ /4\ 2 |-| 3*x - 2 x - 4*x + 5 \1/ ------------ = -------------- + ------------- 2 2 2 x - 4*x + 5 (-x + 2) + 1
/ | | 3*x - 2 | ------------ dx | 2 = | x - 4*x + 5 | /
/ | | 2*x - 4 3* | ------------ dx | 2 / | x - 4*x + 5 | | | 1 / 4* | ------------- dx + -------------------- | 2 2 | (-x + 2) + 1 | /
/ | | 2*x - 4 3* | ------------ dx | 2 | x - 4*x + 5 | / -------------------- 2
2 u = x - 4*x
/ | | 1 3* | ----- du | 5 + u | / 3*log(5 + u) ------------- = ------------ 2 2
/ | | 2*x - 4 3* | ------------ dx | 2 | x - 4*x + 5 | / 2 \ / 3*log\5 + x - 4*x/ -------------------- = ------------------- 2 2
/ | | 1 4* | ------------- dx | 2 | (-x + 2) + 1 | /
v = 2 - x
/ | | 1 4* | ------ dv = 4*atan(v) | 2 | 1 + v | /
/ | | 1 4* | ------------- dx = 4*atan(-2 + x) | 2 | (-x + 2) + 1 | /
/ 2 \ 3*log\5 + x - 4*x/ C + 4*atan(-2 + x) + ------------------- 2
/ | / 2 \ | 3*x - 2 3*log\5 + x - 4*x/ | ------------ dx = C + 4*atan(-2 + x) + ------------------- | 2 2 | x - 4*x + 5 | /
3*log(5) 3*log(2) -pi + 4*atan(2) - -------- + -------- 2 2
=
3*log(5) 3*log(2) -pi + 4*atan(2) - -------- + -------- 2 2
-pi + 4*atan(2) - 3*log(5)/2 + 3*log(2)/2
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.