Archive for April, 2021

Interesting Integral Problem from the JEE Advanced exam

Posted: April 6, 2021 in Mathematics
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Last week I was watching some youtube videos about some Korean undergraduates trying to do the JEE Advanced exam. Apparently this is a highschool exam, an entrance exam for some University or Institution or some sort in India. Some look quite difficult and some are interesting like math contest type questions. I may share them sometimes later. Here is one of the problems.

Divide the numerator and the denominator by (\sqrt{\cos \theta})^5, the integral becomes

\displaystyle I = 3 \int_0^{\pi/2} \frac{\sec^2 \theta}{(1+\sqrt{\tan \theta})^5} \text{ d}\theta

Let

w=1+\sqrt{\tan \theta}, \qquad (w-1)^2 = \tan \theta, \qquad 2(w-1)\text{ d}w = \sec^2 \theta \text{ d}\theta

Then the integral becomes

\displaystyle I = 3 \int_1^{\infty} \frac{2(w-1)\text{ d}w}{w^5} = 6 \int_1^\infty w^{-4}-w^{-5} \text{ d}w

\displaystyle = 6 \left(\frac{w^{-3}}{-3}+\frac{w^{-4}}{4}\right)\bigg|_1^\infty = 6\left(\frac13-\frac14\right) = \frac12