You must have solved the integral of log of cosine

Calculus Level 5

\(I=\displaystyle \int _{ 0 }^{ \pi /2 }{ { x }^{ 2 }\log ( \sec { (x) } ) dx } \)

If the value of \( I \) can be represented as = \(\dfrac{\pi}{A} \zeta{(B)} + \dfrac{{\pi}^{C} \log{D} }{E} \)

Find \(ABCDE+1\)

Details and Assumptions

1) \(A,B,C,D,E\) are positive integers. Also \(ABCDE\) means product of the integers \(A,B,C,D,E\). Also base of \( log \) is \(e\)

2)Remember \(D\) is not divisible by a perfect power(power >1 ) of any integer.

3) \(\displaystyle \zeta{(s)} = \sum _{ n=1 }^{ \infty }{ \frac { 1 }{ { n }^{ s } } } \).

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