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