Til the last breath-2

Calculus Level 4

T=2ϕ(x2+xr=1x(01(3r2y2+4ry+r2)dy))dx{\mathfrak T=\int_{2}^{\color{forestgreen}{\phi}}\left(\dfrac{x^2+x}{\displaystyle\sum_{r=1}^{x}\left(\int_{0}^1\left( 3r^2y^2+4ry+r^2\right)\,dy\right)}\right)\,dx}

T=lnφφ8\Large{\mathfrak{T}=\ln {\dfrac{\color{#EC7300}{\varphi^{\varphi}}}{8}}}

φ=?\huge{ \color{#EC7300}{\varphi}=?}

Given

ϕ=ψ4+1\large\color{forestgreen}{\phi}=\dfrac{\color{#3D99F6}{\psi}}{4}+1

ψ=82168216\large\color{#3D99F6}{\psi}={\sqrt{8\sqrt{216\sqrt{8\sqrt{216\cdots}}}}}


Try the Part-1 here.
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