Was this the cause of yesterday's earthquake?

Calculus Level 4

Mayank and Akul have a very strong feeling that the existence of the following expression was cause of the Afghani earthquake dated 26/10/2015

Let \(f(x)\) = \(\sum _{ r=1 }^{ \infty }{ sec(\frac { x }{ { 2 }^{ r+1 } } ) } .\int _{ 0 }^{ \frac { x }{ { 2 }^{ r+1 } } }{ (1+{ e }^{ sint } } +{ tan }^{ -1 }t)dt.\int _{ -1 }^{ \frac { x }{ { 2 }^{ r+1 } } }{ \frac { ln(1+t) }{ t } } dt\)

for \(x\epsilon [-1,1]\)

Find the value of \(\lim _{ x\xrightarrow { } 0 }{ \frac { f(x) }{ x } } \)

If the answer can be expressed as \(\frac { { \pi }^{ a } }{ b }\)

Find the value of a+b

Remember to be very wise in your approach

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