1 / | | 5*x + 2 | ------------ dx | 2 | x + 2*x + 3 | / 0
Integral((5*x + 2)/(x^2 + 2*x + 3), (x, 0, 1))
/ | | 5*x + 2 | ------------ dx | 2 | x + 2*x + 3 | /
2*x + 2 5*------------ /-3 \ 2 |---| 5*x + 2 x + 2*x + 3 \ 2 / ------------ = -------------- + ------------------------ 2 2 2 x + 2*x + 3 / ___ ___\ |-\/ 2 \/ 2 | |-------*x - -----| + 1 \ 2 2 /
/ | | 5*x + 2 | ------------ dx | 2 = | x + 2*x + 3 | /
/ | | 1 3* | ------------------------ dx / | 2 | | / ___ ___\ | 2*x + 2 | |-\/ 2 \/ 2 | 5* | ------------ dx | |-------*x - -----| + 1 | 2 | \ 2 2 / | x + 2*x + 3 | | / / - -------------------------------- + -------------------- 2 2
/ | | 2*x + 2 5* | ------------ dx | 2 | x + 2*x + 3 | / -------------------- 2
2 u = x + 2*x
/ | | 1 5* | ----- du | 3 + u | / 5*log(3 + u) ------------- = ------------ 2 2
/ | | 2*x + 2 5* | ------------ dx | 2 | x + 2*x + 3 | / 2 \ / 5*log\3 + x + 2*x/ -------------------- = ------------------- 2 2
/ | | 1 -3* | ------------------------ dx | 2 | / ___ ___\ | |-\/ 2 \/ 2 | | |-------*x - -----| + 1 | \ 2 2 / | / --------------------------------- 2
___ ___ \/ 2 x*\/ 2 v = - ----- - ------- 2 2
/ | | 1 -3* | ------ dv | 2 | 1 + v | / -3*atan(v) --------------- = ---------- 2 2
/ | | 1 -3* | ------------------------ dx | 2 | / ___ ___\ | |-\/ 2 \/ 2 | | |-------*x - -----| + 1 / ___ ___\ | \ 2 2 / ___ |\/ 2 x*\/ 2 | | -3*\/ 2 *atan|----- + -------| / \ 2 2 / --------------------------------- = ------------------------------ 2 2
/ ___ ___\ ___ |\/ 2 x*\/ 2 | / 2 \ 3*\/ 2 *atan|----- + -------| 5*log\3 + x + 2*x/ \ 2 2 / C + ------------------- - ----------------------------- 2 2
/ ___ \ / ___ |\/ 2 *(1 + x)| | / 2 \ 3*\/ 2 *atan|-------------| | 5*x + 2 5*log\3 + x + 2*x/ \ 2 / | ------------ dx = C + ------------------- - --------------------------- | 2 2 2 | x + 2*x + 3 | /
/ ___\ ___ |\/ 2 | ___ / ___\ 3*\/ 2 *atan|-----| 5*log(3) 5*log(6) 3*\/ 2 *atan\\/ 2 / \ 2 / - -------- + -------- - ------------------- + ------------------- 2 2 2 2
=
/ ___\ ___ |\/ 2 | ___ / ___\ 3*\/ 2 *atan|-----| 5*log(3) 5*log(6) 3*\/ 2 *atan\\/ 2 / \ 2 / - -------- + -------- - ------------------- + ------------------- 2 2 2 2
-5*log(3)/2 + 5*log(6)/2 - 3*sqrt(2)*atan(sqrt(2))/2 + 3*sqrt(2)*atan(sqrt(2)/2)/2
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.