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Let

a,\ b

be positive real numbers with

a<b

. Define the definite integrals

I_1,\ I_2,\ I_3

I_1=\int_a^b \sin\ (x^2)\ dx,\ I_2=\int_a^b \frac{\cos\ (x^2)}{x^2}\ dx,\ I_3=\int_a^b \frac{\sin\ (x^2)}{x^4}\ dx

.
(1) Find the value of

I_1+\frac 12I_2

in terms of

a,\ b

.
(2) Find the value of

I_2-\frac 32I_3

in terms of

a,\ b

.
(3) For a positive integer

n

, define

K_n=\int_{\sqrt{2n\pi}}^{\sqrt{2(n+1)\pi}} \sin\ (x^2)\ dx+\frac 34\int_{\sqrt{2n\pi}}^{\sqrt{2(n+1)\pi}}\frac{\sin\ (x^2)}{x^4}\ dx

.
Find the value of

\lim_{n\to\infty} 2n\pi \sqrt{2n\pi} K_n

.
2011 Tokyo University of Science entrance exam/Information Sciences, Applied Chemistry, Mechanical Enginerring, Civil Enginerring

Problem Statement