1 / | | 1 | -------- dy | cos(2*y) | / 0
Integral(1/cos(2*y), (y, 0, 1))
/ | | 1 | -------- dy | cos(2*y) | /
1 -------- cos(2*y)
cos(2*y)
1 cos(2*y) -------- = --------- cos(2*y) 2 cos (2*y)
sin(a)^2 + cos(a)^2 = 1
2 2 cos (2*y) = 1 - sin (2*y)
cos(2*y) cos(2*y) --------- = ------------- 2 2 cos (2*y) 1 - sin (2*y)
u = sin(2*y)
/ | | cos(2*y) | ------------- dy | 2 = | 1 - sin (2*y) | /
/ | | cos(2*y) | ------------- dy | 2 = | 1 - sin (2*y) | /
/ | | 1 | ---------- du | / 2\ | 2*\1 - u / | /
1 1 / 1 1 \ ---------- = ---*|----- + -----| / 2\ 2*2 \1 - u 1 + u/ 2*\1 - u /
/ / | | | 1 | 1 | ----- du | ----- du / | 1 + u | 1 - u | | | | 1 / / = | ---------- du = ----------- + ----------- | / 2\ 4 4 | 2*\1 - u / | /
= -log(-1 + u)/4 + log(1 + u)/4
u = sin(2*y)
/ | | 1 log(-1 + sin(2*y)) log(1 + sin(2*y)) | -------- dy = - ------------------ + ----------------- + C0 | cos(2*y) 4 4 | /
/ | | 1 log(-1 + sin(2*y)) log(1 + sin(2*y)) | -------- dy = C - ------------------ + ----------------- | cos(2*y) 4 4 | /
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.