ind the value of constant r such that \( \frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{2 x-3}{x+5}\right)^{4}=\frac{r}{(x+5)^{2}} \)

Find the point(s) at which the function \( f(x)=10-|x| \) equals its average value on the interval \( [-10,10] \) The function equals its average value at \( x=\square \) (Type an integer or a fraction. Use a comma to separate answers as needed.)

Evaluate the following definite integral using the Fundamental Theorem of Calculus. \( \frac{1}{7} \int_{0}^{\ln 4} e^{x} d x \)

Find the following limit or state that it does not exist \[ \lim _{x \rightarrow-1} \frac{4(2 x-2)^{2}-64}{x+1} \]

\( \int _ { - 1 } ^ { 1 } \frac { x \cos ^ { 2 } x } { 1 + x ^ { 2 } } d x \)

\( \int \frac { 3 x ^ { 5 } } { \sqrt { 7 - 12 x ^ { 6 } } } d x = \square \)