Answer to Question #192502 in Real Analysis for Nikhil

Question

State Bonnet's mean value theorem for integrals.

Apply it to show that:

5

| ∫ (cos x/x)dx | ≤ 2/3

3

Answer 1

BONNET MEAN VALUE THEOREM

If both f:[a,b] $\to$ R and g:[a,b]$\to$ R Integrable on [a,b] and f is non negative and monotonically decreasing on [a,b] then there exist a point $\gamma$ $\isin$ [a,b] such that $\int_a^b f(x)g(x)dx=f(a)\int_a^\gamma g(x)dx$

Now we need to show that | $\int_3^5 (cosx/x) dx| \leq \frac 2 3$

Here f(x)=$\frac 1 x$ & g(x) = $cosx$

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 . Here f(a)=f(3)=$\frac 1 3$

So $\int _3^5 (cosx/x)dx = \frac 1 3 \int _3^\gamma (cosx) dx$

=$\frac 1 3 [sin(\gamma)-sin(3)]$

Now | $\int_3^5 (cosx/x) dx | = |\frac 1 3 (sin(\gamma)-sin(3))|$

$\leq \frac 1 3 ( |sin(\gamma)|+|sin(3)|)$

$\leq\frac 1 3 (1 +1)$

$\leq\frac 2 3$ (Proved)