1 / | | t | ------------ dt | 2 | t + 2*t + 4 | / 0
Integral(t/(t^2 + 2*t + 4), (t, 0, 1))
/ | | t | ------------ dt | 2 | t + 2*t + 4 | /
/ 2*t + 2 \ |------------| /-1 \ | 2 | |---| t \t + 2*t + 4/ \ 3 / ------------ = -------------- + ------------------------ 2 2 2 t + 2*t + 4 / ___ ___\ |-\/ 3 \/ 3 | |-------*t - -----| + 1 \ 3 3 /
/ | | t | ------------ dt | 2 = | t + 2*t + 4 | /
/ | | 1 / | ------------------------ dt | | 2 | 2*t + 2 | / ___ ___\ | ------------ dt | |-\/ 3 \/ 3 | | 2 | |-------*t - -----| + 1 | t + 2*t + 4 | \ 3 3 / | | / / ------------------ - ------------------------------ 2 3
/ | | 2*t + 2 | ------------ dt | 2 | t + 2*t + 4 | / ------------------ 2
2 u = t + 2*t
/ | | 1 | ----- du | 4 + u | / log(4 + u) ----------- = ---------- 2 2
/ | | 2*t + 2 | ------------ dt | 2 | t + 2*t + 4 | / 2 \ / log\4 + t + 2*t/ ------------------ = ----------------- 2 2
/ | | 1 - | ------------------------ dt | 2 | / ___ ___\ | |-\/ 3 \/ 3 | | |-------*t - -----| + 1 | \ 3 3 / | / -------------------------------- 3
___ ___ \/ 3 t*\/ 3 v = - ----- - ------- 3 3
/ | | 1 - | ------ dv | 2 | 1 + v | / -atan(v) -------------- = --------- 3 3
/ | | 1 - | ------------------------ dt | 2 | / ___ ___\ | |-\/ 3 \/ 3 | | |-------*t - -----| + 1 / ___ ___\ | \ 3 3 / ___ |\/ 3 t*\/ 3 | | -\/ 3 *atan|----- + -------| / \ 3 3 / -------------------------------- = ----------------------------- 3 3
/ ___ ___\ ___ |\/ 3 t*\/ 3 | / 2 \ \/ 3 *atan|----- + -------| log\4 + t + 2*t/ \ 3 3 / C + ----------------- - --------------------------- 2 3
/ ___ \ / ___ |\/ 3 *(1 + t)| | / 2 \ \/ 3 *atan|-------------| | t log\4 + t + 2*t/ \ 3 / | ------------ dt = C + ----------------- - ------------------------- | 2 2 3 | t + 2*t + 4 | /
/ ___\ ___ |2*\/ 3 | \/ 3 *atan|-------| ___ log(7) log(4) \ 3 / pi*\/ 3 ------ - ------ - ------------------- + -------- 2 2 3 18
=
/ ___\ ___ |2*\/ 3 | \/ 3 *atan|-------| ___ log(7) log(4) \ 3 / pi*\/ 3 ------ - ------ - ------------------- + -------- 2 2 3 18
log(7)/2 - log(4)/2 - sqrt(3)*atan(2*sqrt(3)/3)/3 + pi*sqrt(3)/18
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.