   Chapter 7.P, Problem 7P

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# A function f is defined by f ( x ) = ∫ 0 π cos t cos ( x − t )   d t     0 ≤ x ≤ 2 π Find the minimum value of f.

To determine

To find:

The minimum value of the given function.

Explanation

Given:

f(x)=0πcostcos(xt)dt

Concept:

Integration by substitution. And integrals of familiar trigonometric functions.

Consider f(x)=0πcostcos(xt)dt

The key to note here is that the functions of x inside the integral acts like constants, because integral is with respect to t.

f(x)=0πcostcos(xt)dt=0πcost(cosxcost+sinxsint)dt=cosx0πcos2tdt+sinx0πsintcostdt;In second term substitute sint=u,then du=costdt

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