pi -- 4 / | | 1 | ------------------- dx | 1 + 2*sin(x)*sin(x) | / 0
Integral(1/(1 + (2*sin(x))*sin(x)), (x, 0, pi/4))
/ /x pi\ / /x\ \\ / /x pi\ / /x\ \\ / /x pi\ / /x\ \\ / /x pi\ / /x\ \\ | |- - --| | tan|-| || | |- - --| | tan|-| || _____________ _____________ | |- - --| | tan|-| || _____________ _____________ | |- - --| | tan|-| || | |2 2 | | \2/ || ___ | |2 2 | | \2/ || / ___ / ___ | |2 2 | | \2/ || ___ / ___ / ___ | |2 2 | | \2/ || 5*|pi*floor|------| + atan|----------------|| 2*\/ 6 *|pi*floor|------| + atan|----------------|| 5*\/ 5 - 2*\/ 6 *\/ 5 + 2*\/ 6 *|pi*floor|------| + atan|----------------|| 2*\/ 6 *\/ 5 - 2*\/ 6 *\/ 5 + 2*\/ 6 *|pi*floor|------| + atan|----------------|| / | \ pi / | _____________|| | \ pi / | _____________|| | \ pi / | _____________|| | \ pi / | _____________|| | | | / ___ || | | / ___ || | | / ___ || | | / ___ || | 1 \ \\/ 5 - 2*\/ 6 // \ \\/ 5 - 2*\/ 6 // \ \\/ 5 + 2*\/ 6 // \ \\/ 5 + 2*\/ 6 // | ------------------- dx = C + ----------------------------------------------- + --------------------------------------------------- + ------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------- | 1 + 2*sin(x)*sin(x) _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ | / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / 27*\/ 5 - 2*\/ 6 + 11*\/ 6 *\/ 5 - 2*\/ 6 27*\/ 5 - 2*\/ 6 + 11*\/ 6 *\/ 5 - 2*\/ 6 27*\/ 5 - 2*\/ 6 + 11*\/ 6 *\/ 5 - 2*\/ 6 27*\/ 5 - 2*\/ 6 + 11*\/ 6 *\/ 5 - 2*\/ 6
/ ___ \ _____________ _____________ / / ___ \\ | -1 + \/ 2 | / ___ / ___ | | -1 + \/ 2 || -pi + atan|----------------| \/ 5 - 2*\/ 6 *\/ 5 + 2*\/ 6 *|-pi + atan|----------------|| | _____________| _____________ _____________ | | _____________|| | / ___ | / ___ / ___ | | / ___ || pi \\/ 5 - 2*\/ 6 / pi*\/ 5 - 2*\/ 6 *\/ 5 + 2*\/ 6 \ \\/ 5 + 2*\/ 6 // ------------------------------------------- + ------------------------------------------- + ------------------------------------------- + ---------------------------------------------------------------- _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ 3*\/ 5 - 2*\/ 6 + \/ 6 *\/ 5 - 2*\/ 6 3*\/ 5 - 2*\/ 6 + \/ 6 *\/ 5 - 2*\/ 6 3*\/ 5 - 2*\/ 6 + \/ 6 *\/ 5 - 2*\/ 6 3*\/ 5 - 2*\/ 6 + \/ 6 *\/ 5 - 2*\/ 6
=
/ ___ \ _____________ _____________ / / ___ \\ | -1 + \/ 2 | / ___ / ___ | | -1 + \/ 2 || -pi + atan|----------------| \/ 5 - 2*\/ 6 *\/ 5 + 2*\/ 6 *|-pi + atan|----------------|| | _____________| _____________ _____________ | | _____________|| | / ___ | / ___ / ___ | | / ___ || pi \\/ 5 - 2*\/ 6 / pi*\/ 5 - 2*\/ 6 *\/ 5 + 2*\/ 6 \ \\/ 5 + 2*\/ 6 // ------------------------------------------- + ------------------------------------------- + ------------------------------------------- + ---------------------------------------------------------------- _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ 3*\/ 5 - 2*\/ 6 + \/ 6 *\/ 5 - 2*\/ 6 3*\/ 5 - 2*\/ 6 + \/ 6 *\/ 5 - 2*\/ 6 3*\/ 5 - 2*\/ 6 + \/ 6 *\/ 5 - 2*\/ 6 3*\/ 5 - 2*\/ 6 + \/ 6 *\/ 5 - 2*\/ 6
pi/(3*sqrt(5 - 2*sqrt(6)) + sqrt(6)*sqrt(5 - 2*sqrt(6))) + (-pi + atan((-1 + sqrt(2))/sqrt(5 - 2*sqrt(6))))/(3*sqrt(5 - 2*sqrt(6)) + sqrt(6)*sqrt(5 - 2*sqrt(6))) + pi*sqrt(5 - 2*sqrt(6))*sqrt(5 + 2*sqrt(6))/(3*sqrt(5 - 2*sqrt(6)) + sqrt(6)*sqrt(5 - 2*sqrt(6))) + sqrt(5 - 2*sqrt(6))*sqrt(5 + 2*sqrt(6))*(-pi + atan((-1 + sqrt(2))/sqrt(5 + 2*sqrt(6))))/(3*sqrt(5 - 2*sqrt(6)) + sqrt(6)*sqrt(5 - 2*sqrt(6)))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.