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