How do you use the properties of integrals to verify the inequality #intsinx/x# from pi/4 to pi/2 is less than or equal to #sqrt(2)/2#?
1 Answer
We wish to show that
#int_(pi/4)^(pi/2) sinx/x dx≤ sqrt(2)/2#
We should start by noting that the integral of
The maclaurin series for sine is known to be
To determine the maclaurin series for
#sinx/x = sum_(n = 0)^oo ((-1)^(n - 1)x^(2n))/((2n -1)!) = 1 - x^2/(3!) + x^4/(5!) + ...#
To determine the value of
#int sinx/xdx = sum_(n = 0)^oo ((-1)^(n - 1)x^(2n + 1))/((2n + 1)(2n - 1)!) = x - x^3/(3(3!)) + x^5/(5(5!)) +... + C#
As for the definite integral
#int_(pi/4)^(pi/2) sinx/x dx = pi/2 - (pi/2)^3/18 + (pi/2)^5/600 - (pi/4 - (pi/4)^3/18 + (pi/4)^5/600)#
Even computing the sum of the first few terms, we get
Hopefully this helps!