Integration + Limit

Calculus Level 4

$\large\lim_{a\to \frac\pi2 ^-} \int_0^a \cos x \ln(\cos x) \, dx$

If the integral above can be expressed as $\ln c - b$ , where $b$ and $c$ are positive integers, find $b +c$ .

2 1 5 4 3

1 solution

Rishabh Jain
Mar 28, 2016

Applying IBP:

$\sin x\ln \cos x|_0^a+\int_0^a \dfrac{\sin^2 x dx}{\cos x}$ $= \sin a\ln \cos a +\int_0^a \dfrac{(1-\cos^2 x) dx}{\cos x}$

$=\sin a\ln \cos a+\ln (\sec a+\tan a)-\sin a$

$=\ln ((\cos a)^{\sin a-1}(1+\sin a) )-\sin a$

Applying limits we get

$\lim_{a\to \frac{\pi}{2}^-} (\ln (\cos a)^{\sin a-1}(1+\sin a))-1$

$=\lim_{a\to \frac{\pi}{2}^-} (\ln (\cos a)^{\sin a-1} )+\ln 2-1=\ln 2-1$

Hence $\Large 2+1=\boxed 3$ .

Thanks for the solution.

Rishabh Deep Singh - 5 years, 2 months ago

My pleasure...... :-)

Rishabh Jain - 5 years, 2 months ago

Rishabh in what standard you syudy now

Patel Akshay - 4 years, 3 months ago