1 / | | 3*x + 4 | ------------ dx | 2 | x + 2*x + 5 | / 0
Integral((3*x + 4)/(x^2 + 2*x + 5), (x, 0, 1))
/ | | 3*x + 4 | ------------ dx | 2 | x + 2*x + 5 | /
2*x + 2 3*------------ 2 3*x + 4 x + 2*x + 5 1 ------------ = -------------- + ------------------ 2 2 / 2 \ x + 2*x + 5 |/ x 1\ | 4*||- - - -| + 1| \\ 2 2/ /
/ | | 3*x + 4 | ------------ dx | 2 = | x + 2*x + 5 | /
/ | | 1 / | -------------- dx | | 2 | 2*x + 2 | / x 1\ 3* | ------------ dx | |- - - -| + 1 | 2 | \ 2 2/ | x + 2*x + 5 | | / / -------------------- + -------------------- 4 2
/ | | 2*x + 2 3* | ------------ dx | 2 | x + 2*x + 5 | / -------------------- 2
2 u = x + 2*x
/ | | 1 3* | ----- du | 5 + u | / 3*log(5 + u) ------------- = ------------ 2 2
/ | | 2*x + 2 3* | ------------ dx | 2 | x + 2*x + 5 | / 2 \ / 3*log\5 + x + 2*x/ -------------------- = ------------------- 2 2
/ | | 1 | -------------- dx | 2 | / x 1\ | |- - - -| + 1 | \ 2 2/ | / -------------------- 4
1 x v = - - - - 2 2
/ | | 1 | ------ dv | 2 | 1 + v | / atan(v) ------------ = ------- 4 4
/ | | 1 | -------------- dx | 2 | / x 1\ | |- - - -| + 1 | \ 2 2/ /1 x\ | atan|- + -| / \2 2/ -------------------- = ----------- 4 2
/1 x\ atan|- + -| / 2 \ \2 2/ 3*log\5 + x + 2*x/ C + ----------- + ------------------- 2 2
/ /1 x\ | atan|- + -| / 2 \ | 3*x + 4 \2 2/ 3*log\5 + x + 2*x/ | ------------ dx = C + ----------- + ------------------- | 2 2 2 | x + 2*x + 5 | /
3*log(5) atan(1/2) pi 3*log(8) - -------- - --------- + -- + -------- 2 2 8 2
=
3*log(5) atan(1/2) pi 3*log(8) - -------- - --------- + -- + -------- 2 2 8 2
-3*log(5)/2 - atan(1/2)/2 + pi/8 + 3*log(8)/2
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.