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HMMT 二月 2009 · CALC 赛 · 第 2 题

HMMT February 2009 — CALC Round — Problem 2

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

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