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Question Number 40151 by maxmathsup by imad last updated on 16/Jul/18

l e t F (x) = \int_{0}^{\frac{π}{2}} c o s (x s i n t) d t

1) p r o v e t h a t \forall u \in R 1 - \frac{u^{2}}{2} ⩽ c o s u ⩽ 1 - \frac{u^{2}}{2} + \frac{u^{4}}{24}

2) p r o v e t h a t \frac{π}{2} (1 - \frac{x^{2}}{4}) ⩽ F (x) ⩽ \frac{π}{2} (1 - \frac{x^{2}}{4} + \frac{x^{4}}{64})

Commented by math khazana by abdo last updated on 19/Jul/18

1) w e h a v e c o s (u) = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} u^{2 n}}{(2 n)!}

= 1 - \frac{u^{2}}{2} + \frac{u^{4}}{(4)!} - \frac{u^{6}}{6!} + \dots . \forall u \in R \Rightarrow

1 - \frac{u^{2}}{2} ⩽ c o s u ⩽ 1 - \frac{u^{2}}{2} + \frac{u^{4}}{24}

2) w e g e t 1 - \frac{{(x s i n t)}^{2}}{2} ⩽ c o s (x s i n t) ⩽ 1 - \frac{{(x s i n t)}^{2}}{2}

+ \frac{{(x s i n t)}^{4}}{24} \Rightarrow

\int_{0}^{\frac{π}{2}} (1 - \frac{x^{2}}{2} {s i n}^{2} t) d t ⩽ \int_{0}^{\frac{π}{2}} c o s (x s i n t) d t

⩽ \int_{0}^{\frac{π}{2}} (1 - \frac{x^{2} {s i n}^{2} t}{2} + \frac{x^{4} {s i n}^{4}}{24}) d t b u t

\int_{0}^{\frac{π}{2}} (1 - \frac{x^{2}}{2} {s i n}^{2} t) d t = \frac{π}{2} - \frac{x^{2}}{2} \int_{0}^{\frac{π}{2}} \frac{1 - c o s (2 t)}{2} d t

= \frac{π}{2} - \frac{x^{2}}{4} \frac{π}{2} = \frac{π}{2} (1 - \frac{x^{2}}{4}) a l s o

\int_{0}^{\frac{π}{2}} (1 - \frac{x^{2} {s i n}^{2} t}{2} + \frac{x^{4} {s i n}^{4} t}{24}) d t

= \frac{π}{2} (1 - \frac{x^{2}}{4}) + \frac{x^{4}}{24} \int_{0}^{\frac{π}{2}} {(\frac{1 - c o s (2 t)}{2})}^{2} d t

= \frac{π}{2} (1 - \frac{x^{2}}{4}) + \frac{x^{4}}{96} \int_{0}^{\frac{π}{2}} (1 - 2 c o s (2 t) + {c o s}^{2} (2 t)) d t

= \frac{π}{2} (1 - \frac{x^{2}}{2}) + \frac{x^{4}}{96} \frac{π}{2} + \frac{x^{4}}{96} \int_{0}^{\frac{π}{2}} \frac{1 + c o s (4 t)}{2} d t

= \frac{π}{2} (1 - \frac{x^{2}}{2}) + \frac{π}{2} (\frac{1}{96} + \frac{1}{2.96}) x^{4}

= \frac{π}{2} (1 - \frac{x^{2}}{2}) + \frac{x^{4}}{64} . \frac{π}{2} = \frac{π}{2} (1 - \frac{x^{2}}{2} + \frac{x^{4}}{64}) \Rightarrow

t b e r e s u l t i s p r o v e d .

Answered by tanmay.chaudhury50@gmail.com last updated on 17/Jul/18

p + i q = \int_{0}^{\frac{Π}{2}} e^{i x s i n t} d t

\int_{0}^{\frac{Π}{2}} 1 + i x s i n t + \frac{i^{2} x^{2} {s i n}^{2} t}{2!} + \frac{i^{3} x^{3} {s i n}^{3} t}{3!} + \frac{i^{4} x^{4} {s i n}^{4} t}{4!} + . .

p = F (x) =

\int_{0}^{\frac{Π}{2}} 1 - \frac{x^{2} {s i n}^{2} t}{4} + \frac{x^{4} {s i n}^{4} t}{24} + . . d t

1 ⩾ {s i n}^{2} t ⩾ 0 \dots 1 ⩾ {s i n}^{2 k} t ⩾ 0 2 k = e v e n

s o \int_{0}^{\frac{Π}{2}} 1 - \frac{x^{2}}{4} d t ⩽ F (x) ⩽ \int_{0}^{\frac{Π}{2}} 1 - \frac{x^{2}}{4} + \frac{x^{4}}{24} d t

(1 - \frac{x^{2}}{4}) \frac{Π}{2} ⩽ F (x) ⩽ (1 - \frac{x^{2}}{4} + \frac{x^{4}}{24}) \frac{Π}{2}