1 / | | x | ---------- dx | 2 | x + x + 1 | / 0
Integral(x/(x^2 + x + 1), (x, 0, 1))
/ | | x | 1*---------- dx | 2 | x + x + 1 | /
/ 1*2*x + 1 \ |--------------| / -1 \ | 2 | |-----| x \1*x + 1*x + 1/ \2*3/4/ ---------- = ---------------- + ------------------------- 2 2 2 x + x + 1 / ___ ___\ |-2*\/ 3 \/ 3 | |--------*x - -----| + 1 \ 3 3 /
/ | | x | 1*---------- dx | 2 = | x + x + 1 | /
/ | | 1 / 2* | ------------------------- dx | | 2 | 1*2*x + 1 | / ___ ___\ | -------------- dx | |-2*\/ 3 \/ 3 | | 2 | |--------*x - -----| + 1 | 1*x + 1*x + 1 | \ 3 3 / | | / / -------------------- - --------------------------------- 2 3
/ | | 1*2*x + 1 | -------------- dx | 2 | 1*x + 1*x + 1 | / -------------------- 2
2 u = x + x
/ | | 1 | ----- du | 1 + u | / log(1 + u) ----------- = ---------- 2 2
/ | | 1*2*x + 1 | -------------- dx | 2 | 1*x + 1*x + 1 | / 2\ / log\1 + x + x / -------------------- = --------------- 2 2
/ | | 1 -2* | ------------------------- dx | 2 | / ___ ___\ | |-2*\/ 3 \/ 3 | | |--------*x - -----| + 1 | \ 3 3 / | / ---------------------------------- 3
___ ___ \/ 3 2*x*\/ 3 v = - ----- - --------- 3 3
/ | | 1 -2* | ------ dv | 2 | 1 + v | / -2*atan(v) --------------- = ---------- 3 3
/ | | 1 -2* | ------------------------- dx | 2 | / ___ ___\ | |-2*\/ 3 \/ 3 | | |--------*x - -----| + 1 / ___ ___\ | \ 3 3 / ___ |\/ 3 2*x*\/ 3 | | -\/ 3 *atan|----- + ---------| / \ 3 3 / ---------------------------------- = ------------------------------- 3 3
/ ___ ___\ ___ |\/ 3 2*x*\/ 3 | / 2\ \/ 3 *atan|----- + ---------| log\1 + x + x / \ 3 3 / C + --------------- - ----------------------------- 2 3
/ ___ \ / ___ |2*\/ 3 *(1/2 + x)| | / 2\ \/ 3 *atan|-----------------| | x log\1 + x + x / \ 3 / | ---------- dx = C + --------------- - ----------------------------- | 2 2 3 | x + x + 1 | /
___ log(3) pi*\/ 3 ------ - -------- 2 18
=
___ log(3) pi*\/ 3 ------ - -------- 2 18
Use the examples entering the upper and lower limits of integration.