An Interesting Integral

Calculus Level 5

$\displaystyle \int_0^{\pi/3}\ln^2\left(\dfrac{\sin{x}}{\sin\left(x+\pi/3\right)}\right)\,dx$

The given integral is equal to $\dfrac{A}{B}\pi^C$ for integers $A, B, C$ where $A$ is a prime number. Submit your answer as the value of $A+B$ .

The answer is 86.

2 solutions

Mark Hennings
Nov 9, 2018

This appeared as part of Season 2 of the Brilliant Integration Contest . The integral is $\tfrac{5}{81}\pi^3$ .

Aaghaz Mahajan
Nov 8, 2018

Note: This is not a solution......

@Digvijay Singh Hey....why did you change the question??? The earlier version was better since the usage of polar coordinates led to the same thing after some manipulations....!! Btw, could you please tell me, where do you get these AMAZING questions from??? For instance, I felt so bad after doing the question regarding locus of semicircular arcs wrong......I knew the equation but didn't simplify the integral.......!!

@Digvijay Singh Please do suggest some books........ Btw I have finished Stewart's Calculus which you had suggested me a long time ago....

Aaghaz Mahajan - 2 years, 7 months ago

For pleasure reading, you can learn some cool integral tricks and advanced integrals from this book: Inside Interesting Integrals

Digvijay Singh - 2 years, 7 months ago

OK thanks!!!!

Aaghaz Mahajan - 2 years, 7 months ago

I changed the question because a lot of people couldn't solve it. I felt that it was a little too complicated.

I make the questions myself. Of course a lot of the stuff is borrowed/inspired from different sources like mathcurve.com . For example, the locus problem was inspired from here .

Digvijay Singh - 2 years, 7 months ago

Ohh I see!!! I know about that site!!! I got to know about the "Elastica" curve from there!!!! But it is really god if you create these problems on your own!!! Are you in college??

Aaghaz Mahajan - 2 years, 7 months ago

An Interesting Integral

The answer is 86.

2 solutions

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