1 / | | x | ------------ dx | 2 | x + 4*x + 5 | / 0
Integral(x/(x^2 + 4*x + 5), (x, 0, 1))
/ | | x | 1*------------ dx | 2 | x + 4*x + 5 | /
/ 1*2*x + 4 \ |--------------| /-2 \ | 2 | |---| x \1*x + 4*x + 5/ \ 1 / ------------ = ---------------- + ------------- 2 2 2 x + 4*x + 5 (-x - 2) + 1
/ | | x | 1*------------ dx | 2 = | x + 4*x + 5 | /
/ | | 1*2*x + 4 | -------------- dx | 2 | 1*x + 4*x + 5 / | | / | 1 -------------------- - 2* | ------------- dx 2 | 2 | (-x - 2) + 1 | /
/ | | 1*2*x + 4 | -------------- dx | 2 | 1*x + 4*x + 5 | / -------------------- 2
2 u = x + 4*x
/ | | 1 | ----- du | 5 + u | / log(5 + u) ----------- = ---------- 2 2
/ | | 1*2*x + 4 | -------------- dx | 2 | 1*x + 4*x + 5 | / 2 \ / log\5 + x + 4*x/ -------------------- = ----------------- 2 2
/ | | 1 -2* | ------------- dx | 2 | (-x - 2) + 1 | /
v = -2 - x
/ | | 1 -2* | ------ dv = -2*atan(v) | 2 | 1 + v | /
/ | | 1 -2* | ------------- dx = -2*atan(2 + x) | 2 | (-x - 2) + 1 | /
/ 2 \ log\5 + x + 4*x/ C + ----------------- - 2*atan(2 + x) 2
/ | / 2 \ | x log\5 + x + 4*x/ | ------------ dx = C + ----------------- - 2*atan(2 + x) | 2 2 | x + 4*x + 5 | /
log(10) log(5) ------- - 2*atan(3) + 2*atan(2) - ------ 2 2
=
log(10) log(5) ------- - 2*atan(3) + 2*atan(2) - ------ 2 2
Use the examples entering the upper and lower limits of integration.