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Calculus Level 4

∫ 0 2 π 5 s i n ( x ⌊ x ⌋ ) d x F i n d t h e v a l u e o f t h e a b o v e i n t e g r a l . I f S i s t h e g i v e n v a l u e f i n d ⌊ 100 S ⌋ . \int _{ 0 }^{ \frac { 2\pi }{ 5 } }{ sin\left( { x }^{ \left\lfloor x \right\rfloor } \right) } dx\\ Find\quad the\quad value\quad of\quad the\quad above\\ integral.\quad If\quad S\quad is\quad the\quad given\quad value\\ find\quad \left\lfloor 100S \right\rfloor .\\ \\ ⌊ x ⌋ r e p r e s e n t s t h e g r e a t e s t i n t e g e r l e s s t h a n o r e q u a l t o x . \left\lfloor x \right\rfloor \quad represents\quad the\quad greatest\quad integer\\ less\quad than\quad or\quad equal\quad to\quad x.


The answer is 107.

Rishi Sharma by Rishi Sharma , 32, India

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1 solution

Aaaaaa Bbbbbb
Sep 5, 2015

The integral can be divided into two components: S = ∫ 0 2 π 5 s i n ( x ⌊ x ⌋ ) d x = ∫ 0 1 s i n ( 1 ) d x + ∫ 1 2 π 5 s i n ( x ) d x = 1.072 S=\int_0^{\frac{2\pi}{5}} sin(x^{\lfloor x \rfloor}) dx = \int_0^1 sin(1) dx + \int_1^{\frac{2\pi}{5}} sin(x) dx = \boxed{1.072} ⌊ 100 S ⌋ = 107 \lfloor 100S \rfloor = \boxed{107}

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