MathDB

(a)

Compute - \begin{align*} \frac{\mathrm{d}}{\mathrm{d}x} \bigg[ \int_{0}^{e^x} \log ( t ) \cos^4 ( t ) \mathrm{d}t \bigg] \end{align*}

(b)

For

x > 0

define

F ( x ) = \int_{1}^{x} t \log ( t ) \mathrm{d}t .

\\ \\

1.

Determine the open interval(s) (if any) where

F ( x )

is decreasing and all the open interval(s) (if any) where

F ( x )

is increasing.\\ \\

2.

Determine all the local minima of

F ( x )

(if any) and all the local maxima of

F ( x )

(if any)

.

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