1 / | | 1 | ------ dx | cos(x) | / 0
Integral(1/cos(x), (x, 0, 1))
/ | | 1 | ------ dx | cos(x) | /
1 ------ cos(x)
cos(x)
1 cos(x) ------ = ------- cos(x) 2 cos (x)
sin(a)^2 + cos(a)^2 = 1
2 2 cos (x) = 1 - sin (x)
cos(x) cos(x) ------- = ----------- 2 2 cos (x) 1 - sin (x)
u = sin(x)
/ | | cos(x) | ----------- dx | 2 = | 1 - sin (x) | /
/ | | cos(x) | ----------- dx | 2 = | 1 - sin (x) | /
/ | | 1 | ------ du | 2 | 1 - u | /
1 1 ----- + ----- 1 1 - u 1 + u ------ = ------------- 2 2 1 - u
/ / | | | 1 | 1 | ----- du | ----- du / | 1 + u | 1 - u | | | | 1 / / = | ------ du = ----------- + ----------- | 2 2 2 | 1 - u | /
= log(1 + u)/2 - log(-1 + u)/2
u = sin(x)
/ | | 1 log(1 + sin(x)) log(-1 + sin(x)) | ------ dx = --------------- - ---------------- + C0 | cos(x) 2 2 | /
/ | | 1 log(1 + sin(x)) log(-1 + sin(x)) | ------ dx = C + --------------- - ---------------- | cos(x) 2 2 | /
log(1 + sin(1)) log(1 - sin(1)) --------------- - --------------- 2 2
=
log(1 + sin(1)) log(1 - sin(1)) --------------- - --------------- 2 2
log(1 + sin(1))/2 - log(1 - sin(1))/2
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.