91Ó°ÊÓ

Finding an Indefinite Integral of a Trigonometric Function In Exercises \(31-40\) , find the indefinite integral. $$ \int \frac{9}{5-4 x} d x $$

Verifying Properties of Logarithms In Exercises 35 and \(36,(a)\) verify that \(f=g\) by using a graphing utility to graph \(f\) and \(g\) in the same viewing window and (b) verify that \(f=g\) algebraically. $$ f(x)=\ln \sqrt{x\left(x^{2}+1\right)}, \quad g(x)=\frac{1}{2}\left[\ln x+\ln \left(x^{2}+1\right)\right] $$

Verifying Inverse Functions In Exercises 35 and 36 , illustrate that the functions are inverse functions of each other by sketching their graphs on the same set of coordinate axes. $$ \begin{array}{l}{f(x)=3^{x}} \\ {g(x)=\log _{3} x}\end{array} $$

Finding a Derivative In Exercises \(33-54,\) find the derivative. $$ y=e^{-2 x^{3}} $$

Completing the Square In Exercises \(33-42,\) find or evaluate the integral by completing the square. $$ \int \frac{2 x-5}{x^{2}+2 x+2} d x $$

In Exercises 33–36, find an equation of the tangent line to the graph of the function at the given point. $$ y=e^{\sinh x}, \quad(0,1) $$

In Exercises 37–40, find any relative extrema of the function. Use a graphing utility to confirm your result. $$ f(x)=\sin x \sinh x-\cos x \cosh x, \quad-4 \leq x \leq 4 $$

Finding a Derivative In Exercises \(37-58\) , find the derivative of the function. (Hint: In some exercises, you may find it helpful to apply logarithmic properties before differentiating.) $$ f(x)=4^{x} $$

Finding a Derivative In Exercises \(33-54,\) find the derivative. $$ y=e^{x-4} $$

Completing the Square In Exercises \(33-42,\) find or evaluate the integral by completing the square. $$ \int \frac{1}{\sqrt{-x^{2}-4 x}} d x $$