Dueling functions

Calculus Level 4

\(f\left( x \right) =\int _{ x }^{ { x }^{ 2 } }{ \left( { t }^{ 2 }+3 \right) } d\left( g\left( t \right) \right) \\ g\left( x \right) ={ x }^{ 2 }\ln { x }\)

Given the following composition of functions, \(x\) can be expressed in the form \({ e }^{ -d }\) for \(x>0\). Under which real values of \(d\) is \(f\left( x \right)\) decreasing?

Details and Assumptions

\(e\) denotes Euler's number, the base of the natural logarithm.


This problem is original.

Picture credits: F-16 Releases Four Flares, Wikipedia

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