1 / | | t + 2 | ------------ dt | 2 | t + 2*t + 5 | / 0
Integral((t + 2)/(t^2 + 2*t + 5), (t, 0, 1))
/ | | t + 2 | ------------ dt | 2 | t + 2*t + 5 | /
/ 2*t + 2 \ |------------| | 2 | t + 2 \t + 2*t + 5/ 1 ------------ = -------------- + ------------------ 2 2 / 2 \ t + 2*t + 5 |/ t 1\ | 4*||- - - -| + 1| \\ 2 2/ /
/ | | t + 2 | ------------ dt | 2 = | t + 2*t + 5 | /
/ | / | 1 | | -------------- dt | 2*t + 2 | 2 | ------------ dt | / t 1\ | 2 | |- - - -| + 1 | t + 2*t + 5 | \ 2 2/ | | / / ------------------ + -------------------- 2 4
/ | | 2*t + 2 | ------------ dt | 2 | t + 2*t + 5 | / ------------------ 2
2 u = t + 2*t
/ | | 1 | ----- du | 5 + u | / log(5 + u) ----------- = ---------- 2 2
/ | | 2*t + 2 | ------------ dt | 2 | t + 2*t + 5 | / 2 \ / log\5 + t + 2*t/ ------------------ = ----------------- 2 2
/ | | 1 | -------------- dt | 2 | / t 1\ | |- - - -| + 1 | \ 2 2/ | / -------------------- 4
1 t v = - - - - 2 2
/ | | 1 | ------ dv | 2 | 1 + v | / atan(v) ------------ = ------- 4 4
/ | | 1 | -------------- dt | 2 | / t 1\ | |- - - -| + 1 | \ 2 2/ /1 t\ | atan|- + -| / \2 2/ -------------------- = ----------- 4 2
/1 t\ atan|- + -| / 2 \ \2 2/ log\5 + t + 2*t/ C + ----------- + ----------------- 2 2
/ /1 t\ | atan|- + -| / 2 \ | t + 2 \2 2/ log\5 + t + 2*t/ | ------------ dt = C + ----------- + ----------------- | 2 2 2 | t + 2*t + 5 | /
log(8) atan(1/2) log(5) pi ------ - --------- - ------ + -- 2 2 2 8
=
log(8) atan(1/2) log(5) pi ------ - --------- - ------ + -- 2 2 2 8
log(8)/2 - atan(1/2)/2 - log(5)/2 + pi/8
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.