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  • 1 # 大寶8211

    首先求∫sec^3(x) dx:記I=∫sec^3(x) dx,則I

    =∫sec(x)*sec^2(x) dx

    =∫sec(x)*[tan(x)]" dx

    =sec(x)*tan(x)-∫[sec(x)]"*tan(x) dx

    =sec(x)*tan(x)-∫[sec(x)*tan(x)]*tan(x) dx

    =sec(x)*tan(x)-∫sec(x)*tan^2(x) dx

    =sec(x)*tan(x)-∫sec(x)*[sec^2(x)-1] dx

    =sec(x)*tan(x)-∫sec^3(x) dx+∫sec(x) dx

    =sec(x)*tan(x)-I+ln|sec(x)+tan(x)|+C,

    所以2I=sec(x)*tan(x)+ln|sec(x)+tan(x)|+C,

    I=sec(x)*tan(x)/2+ln|sec(x)+tan(x)|/2+C,C為任意常數

    然後求∫sec^5(x) dx:記J=∫sec^5(x) dx,則J

    =∫sec^3(x)*sec^2(x) dx

    =∫sec^3(x)*[tan(x)]" dx

    =sec^3(x)*tan(x)-∫[sec^3(x)]"*tan(x) dx

    =sec^3(x)*tan(x)-∫3sec^2(x)*[sec(x)*tan(x)]*tan(x) dx

    =sec^3(x)*tan(x)-3∫sec^3(x)*tan^2(x) dx

    =sec^3(x)*tan(x)-3∫sec^3(x)*[sec^2(x)-1] dx

    =sec^3(x)*tan(x)-3∫sec^5(x) dx+3∫sec^3(x) dx

    =sec^3(x)*tan(x)-3J+3I,

    所以4J=sec^3(x)*tan(x)+3I,

    J=sec^3(x)*tan(x)/4+3I/4

    =sec^3(x)*tan(x)/4+3sec(x)*tan(x)/8+3ln|sec(x)+tan(x)|/8+C,

    C為任意常數

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