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HMMT 二月 2009 · CALC 赛 · 第 2 题

HMMT February 2009 — CALC Round — Problem 2

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 3 ] The differentiable function F : R → R satisfies F (0) = − 1 andd
    F ( x ) = sin(sin(sin(sin( x )))) · cos(sin(sin( x ))) · cos(sin( x )) · cos( x ) .
    Find F ( x ) as a function of x .dx A
解析

英文解析

  1. [ 3 ] The differentiable function F : R → R satisfies F (0) = − 1 andd
    F ( x ) = sin(sin(sin(sin( x )))) · cos(sin(sin( x ))) · cos(sin( x )) · cos( x ) .
    Find F ( x ) as a function of x .dx
    Answer: − cos(sin(sin(sin( x ))))
    Solution: Substituting u = sin(sin(sin( x ))), we find
    ∫
    F ( x ) = sin( u ) du = − cos( u ) + C.
    for some C . Since F (0) = 1 we find C = 0. A