# Calcgebra

Calculus Level 5

First, compute

$\mathfrak{S}= \displaystyle \sum_{e=1}^{5} \sum_{d=1}^{e} \sum_{c=1}^{d} \sum_{b=1}^{c} \sum_{a=1}^{b} 1$.

If the above sum $$\mathfrak{S} =\zeta ^{3} + 1$$, with $$\zeta \in \mathbb {N}$$ then evaluate

$\mathfrak{I} = \displaystyle \int_{0}^{\pi/4} \tan^{\zeta^{3}} x \sec^{4} x \, dx$

Finally, if $$\mathfrak{I}$$ can be represented in the simple form $$\dfrac{a}{b}$$, where $$a, b$$ are coprime positive integers, find $$a+b$$.

Nota Bene: NO CALCULATORS!

Bonus: Can you find a way to compute $$\mathfrak{S}$$ efficiently?

Inspiration

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