0 / | | a*x | ------- dx | 2 | cos (x) | / pi -- 4
Integral((a*x)/cos(x)^2, (x, pi/4, 0))
/ / / 2/x\\ / /x\\ / /x\\ 2/x\ / /x\\ 2/x\ / /x\\ 2/x\ / 2/x\\ /x\ \ | |log|1 + tan |-|| log|1 + tan|-|| log|-1 + tan|-|| tan |-|*log|1 + tan|-|| tan |-|*log|-1 + tan|-|| tan |-|*log|1 + tan |-|| 2*x*tan|-| | | a*x | \ \2// \ \2// \ \2// \2/ \ \2// \2/ \ \2// \2/ \ \2// \2/ | | ------- dx = C + a*|---------------- - --------------- - ---------------- + ----------------------- + ------------------------ - ------------------------ - ------------| | 2 | 2/x\ 2/x\ 2/x\ 2/x\ 2/x\ 2/x\ 2/x\| | cos (x) | -1 + tan |-| -1 + tan |-| -1 + tan |-| -1 + tan |-| -1 + tan |-| -1 + tan |-| -1 + tan |-|| | \ \2/ \2/ \2/ \2/ \2/ \2/ \2// /
/ / 2\ 2 2 2 / 2\ \ | | / ___\ | / ___\ / ___\ / ___\ / / ___\\ / ___\ / ___\ / ___\ | / ___\ | / ___\ | |log\1 + \-1 + \/ 2 / / pi*I + log\2 - \/ 2 / log\\/ 2 / \-1 + \/ 2 / *\pi*I + log\2 - \/ 2 // \-1 + \/ 2 / *log\\/ 2 / \-1 + \/ 2 / *log\1 + \-1 + \/ 2 / / pi*\-1 + \/ 2 / | - a*|---------------------- - --------------------- - ------------------ + ------------------------------------- + ------------------------ - ------------------------------------ - ----------------------| + pi*I*a | 2 2 2 2 2 2 / 2\| | / ___\ / ___\ / ___\ / ___\ / ___\ / ___\ | / ___\ || \ -1 + \-1 + \/ 2 / -1 + \-1 + \/ 2 / -1 + \-1 + \/ 2 / -1 + \-1 + \/ 2 / -1 + \-1 + \/ 2 / -1 + \-1 + \/ 2 / 2*\-1 + \-1 + \/ 2 / //
=
/ / 2\ 2 2 2 / 2\ \ | | / ___\ | / ___\ / ___\ / ___\ / / ___\\ / ___\ / ___\ / ___\ | / ___\ | / ___\ | |log\1 + \-1 + \/ 2 / / pi*I + log\2 - \/ 2 / log\\/ 2 / \-1 + \/ 2 / *\pi*I + log\2 - \/ 2 // \-1 + \/ 2 / *log\\/ 2 / \-1 + \/ 2 / *log\1 + \-1 + \/ 2 / / pi*\-1 + \/ 2 / | - a*|---------------------- - --------------------- - ------------------ + ------------------------------------- + ------------------------ - ------------------------------------ - ----------------------| + pi*I*a | 2 2 2 2 2 2 / 2\| | / ___\ / ___\ / ___\ / ___\ / ___\ / ___\ | / ___\ || \ -1 + \-1 + \/ 2 / -1 + \-1 + \/ 2 / -1 + \-1 + \/ 2 / -1 + \-1 + \/ 2 / -1 + \-1 + \/ 2 / -1 + \-1 + \/ 2 / 2*\-1 + \-1 + \/ 2 / //
-a*(log(1 + (-1 + sqrt(2))^2)/(-1 + (-1 + sqrt(2))^2) - (pi*i + log(2 - sqrt(2)))/(-1 + (-1 + sqrt(2))^2) - log(sqrt(2))/(-1 + (-1 + sqrt(2))^2) + (-1 + sqrt(2))^2*(pi*i + log(2 - sqrt(2)))/(-1 + (-1 + sqrt(2))^2) + (-1 + sqrt(2))^2*log(sqrt(2))/(-1 + (-1 + sqrt(2))^2) - (-1 + sqrt(2))^2*log(1 + (-1 + sqrt(2))^2)/(-1 + (-1 + sqrt(2))^2) - pi*(-1 + sqrt(2))/(2*(-1 + (-1 + sqrt(2))^2))) + pi*i*a
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.