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how you show integral sign(0-pie/2)f(x)dx < or equal 1? given x*f(x)=sinx

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mayamorosz | Student, College Freshman | eNoter

Posted October 9, 2012 at 1:23 PM via web

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how you show integral sign(0-pie/2)f(x)dx < or equal 1?

given x*f(x)=sinx

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted December 25, 2012 at 6:18 PM (Answer #1)

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You need to prove the following inequality such that:

`int_0^(pi/2) f(x) dx <= cos 1`

The problem provides the information that `x*f(x) = sin x => f(x) = sin x/x.`

Since `sin x/x < sin x` , if `x in (0,pi/2), ` hence `int_0^(pi/2) f(x) dx = int_0^(pi/2)sin x/x dx < int_0^(pi/2) sin x dx`

Evaluating `int_0^(pi/2) sin x dx`  yields:

`int_0^(pi/2) sin x dx = -cos x|_0^(pi/2)`

You need to use fundamental theorem of calculus, such that:

`int_0^(pi/2) sin x dx = - cos (pi/2) - (- cos 0)`

`int_0^(pi/2) sin x dx = -0 - (-1) = 1`

Hence, evaluating `int_0^(pi/2) sin x dx = 1` , yields that the given inequality `int_0^(pi/2) f(x) dx = int_0^(pi/2) sin x/x dx < int_0^(pi/2) sin x dx = 1`  holds.

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