1 / | | 3 | -------- dx | cos(2*x) | / 0
Integral(3/cos(2*x), (x, 0, 1))
/ | | 3 | -------- dx | cos(2*x) | /
3 -------- cos(2*x)
cos(2*x)
3 3*cos(2*x) -------- = ---------- cos(2*x) 2 cos (2*x)
sin(a)^2 + cos(a)^2 = 1
2 2 cos (2*x) = 1 - sin (2*x)
3*cos(2*x) 3*cos(2*x) ---------- = ------------- 2 2 cos (2*x) 1 - sin (2*x)
u = sin(2*x)
/ | | 3*cos(2*x) | ------------- dx | 2 = | 1 - sin (2*x) | /
/ | | 3*cos(2*x) | ------------- dx | 2 = | 1 - sin (2*x) | /
/ | | 3 | ---------- du | / 2\ | 2*\1 - u / | /
/3\ |-| 3 \2/ / 1 1 \ ---------- = ---*|----- + -----| / 2\ 2 \1 - u 1 + u/ 2*\1 - u /
/ / | | | 1 | 1 3* | ----- du 3* | ----- du / | 1 + u | 1 - u | | | | 3 / / = | ---------- du = ------------- + ------------- | / 2\ 4 4 | 2*\1 - u / | /
= -3*log(-1 + u)/4 + 3*log(1 + u)/4
u = sin(2*x)
/ | | 3 3*log(-1 + sin(2*x)) 3*log(1 + sin(2*x)) | -------- dx = - -------------------- + ------------------- + C0 | cos(2*x) 4 4 | /
/ | | 3 3*log(-1 + sin(2*x)) 3*log(1 + sin(2*x)) | -------- dx = C - -------------------- + ------------------- | cos(2*x) 4 4 | /
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.