# 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.

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