Answer to Question #203273 in Real Analysis for Raj kumar

Question

State Bonnet’s mean value theorem for integrals. Apply it to show that:

|₃∫⁵ cosxdx/x|≤ 2/3

Answer 1

BONNET MEAN VALUE THEOREM

If both f:[a,b] → R and g:[a,b]→ R Integrable on [a,b] and f is non negative and monotonically decreasing on [a,b] then there exist a point n ∈ [a,b] such that

$\int$^a_b f(x) g(x) dx = f(a) $\int$ⁿ_a g(x) dx

To prove the above indefinite integral,

Let f(x) = 1/x and g(x) = cos x

Here both f(x) and g(x) are integrable on [3,5]

and f(x) is non negative and monotonic decreasing on [3,5] .

So bonnet theorom is applicable here .

Now f(a) = f(3) = 1/3

$\int$⁵₃ cosx/x dx = 1/3 $\int$ⁿ₃ cos x dx

= 1/3 ( sin n - sin 3)

Maximum value of sine function is 1

So (1+1)/3 = 2/3

Hence proved