HMMT 二月 2007 · 冲刺赛 · 第 33 题

英文解析

33. [ 18 ] Compute
    ∫
    9 x + 42
    dx.
    5 2
    x + 3 x + x
    (No, your TI-89 doesn’t know how to do this one. Yes, the end is near.)1
    Answer: ln . We break the given integral into two pieces:80
    ∫ ∫ ∫23
    2 2 2
    4 4
    9 x + 4 x + 3 x + 1 5 x + 6 x + 1
    dx = 5 dx − dx.
    5 2 5 2 5 2
    x + 3 x + x x + 3 x + x x + 3 x + x
    1 1 1
    These two new integrals are easily computed; for, the first integrand reduces to 1 /x and the second is 11
    ′
    of the form f ( x ) /f ( x ). We obtain
    [ ]
    5 2280
    5 ln | x | − ln | x + 3 x + x | = ln 32 − ln 46 + ln 5 = ln
    5 2 4231
    Motivation. Writing f ( x ) = 9 x + 4 and g ( x ) = x + 3 x + x = x ( x + 3 x + 1), we wish to findthe antiderivative of f ( x ) /g ( x ). It makes sense to consider other rational functions with denominatorg ( x ) that have an exact antiderivative. Clearly, if the numerator were f ( x ) = x + 3 x + 1 or a 4
    constant multiple, then we can integrate the function. Another trivial case is if the numerator were 1
    ′ 4
    f ( x ) = g ( x ) = 5 x + 6 x + 1 or a constant multiple. Guessing that f ( x ) is a linear combination off ( x ) and f ( x ), we easily find that f ( x ) = 9 x + 4 = 5 f ( x ) − f ( x ).2
    1 2 1 2
    10 HARVARD-MIT MATHEMATICS TOURNAMENT, 24 FEBRUARY 2007 — GUTS ROUNDth