# 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?