38. Use integration formulas to solve each integral. You may have to use algebra, educated guess-and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. (a) (10 points) \( \int\left(x^{2}-3 x^{8}-7\right) d x= \) (b) (10 points) \( \int 3 \cos (x) d x= \) (c) (10 points) \( \int x^{2}\left(x^{3}+1\right) d x= \)

Find the intervals on which \( f \) is increasing and the intervals on which it is decreasing. \( f(x)=\frac{x^{4}}{4}-3 x^{3}+9 x^{2} \)

Given \( f(x, y)=3 x^{3}+x y^{2}-2 y^{6} \), find \( f_{y} \) \( 2 x-60 y \) \( 18 x \) \( 2 y \) None of these answers

Determine the intervals on which the following function is concave up or concave down. Identify any inflection points. \( f(x)=x^{4}-3 x^{3}+9 \)

Find the derivative of the following function. \( g(t)=4 \sqrt{t} \) \( g^{\prime}(t)=\square \)

Find the derivative of the following function by first expanding the expression. \[ f(x)=(6 x+1)\left(5 x^{2}+7\right) \] \( f^{\prime}(x)=\square \)