1 / | | x | ------------ dx | 2 | x + 2*x + 4 | / 0
/ | | x | 1*------------ dx | 2 | x + 2*x + 4 | /
/ 1*2*x + 2 \ |--------------| /-1 \ | 2 | |---| x \1*x + 2*x + 4/ \ 3 / ------------ = ---------------- + ------------------------ 2 2 2 x + 2*x + 4 / ___ ___\ |-\/ 3 \/ 3 | |-------*x - -----| + 1 \ 3 3 /
/ | | x | 1*------------ dx | 2 = | x + 2*x + 4 | /
/ | | 1 / | ------------------------ dx | | 2 | 1*2*x + 2 | / ___ ___\ | -------------- dx | |-\/ 3 \/ 3 | | 2 | |-------*x - -----| + 1 | 1*x + 2*x + 4 | \ 3 3 / | | / / -------------------- - ------------------------------ 2 3
/ | | 1*2*x + 2 | -------------- dx | 2 | 1*x + 2*x + 4 | / -------------------- 2
2 u = x + 2*x
/ | | 1 | ----- du | 4 + u | / log(4 + u) ----------- = ---------- 2 2
/ | | 1*2*x + 2 | -------------- dx | 2 | 1*x + 2*x + 4 | / 2 \ / log\4 + x + 2*x/ -------------------- = ----------------- 2 2
/ | | 1 - | ------------------------ dx | 2 | / ___ ___\ | |-\/ 3 \/ 3 | | |-------*x - -----| + 1 | \ 3 3 / | / -------------------------------- 3
___ ___ \/ 3 x*\/ 3 v = - ----- - ------- 3 3
/ | | 1 - | ------ dv | 2 | 1 + v | / -atan(v) -------------- = --------- 3 3
/ | | 1 - | ------------------------ dx | 2 | / ___ ___\ | |-\/ 3 \/ 3 | | |-------*x - -----| + 1 / ___ ___\ | \ 3 3 / ___ |\/ 3 x*\/ 3 | | -\/ 3 *atan|----- + -------| / \ 3 3 / -------------------------------- = ----------------------------- 3 3
/ ___ ___\ ___ |\/ 3 x*\/ 3 | / 2 \ \/ 3 *atan|----- + -------| log\4 + x + 2*x/ \ 3 3 / C + ----------------- - --------------------------- 2 3
/ ___ \ / ___ |\/ 3 *(1 + x)| | / 2 \ \/ 3 *atan|-------------| | x log\4 + x + 2*x/ \ 3 / | ------------ dx = C + ----------------- - ------------------------- | 2 2 3 | x + 2*x + 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
Данные примеры также можно применять при вводе верхнего и нижнего предела интегрирования.