91Ó°ÊÓ

State (if possible) the method or integration formula you would use to find the antiderivative. Explain why you chose that method or formula. Do not integrate. $$ \int \frac{e^{x}}{e^{2 x}+1} d x $$

Find any asymptotes and relative extrema that may exist and use a graphing utility to graph the function. (Hint: Some of the limits required in finding asymptotes have been found in preceding exercises.) $$ y=2 x e^{-x} $$

In Exercises \(73-78\), use a computer algebra system to find the integral. Graph the antiderivatives for two different values of the constant of integration. \(\int \cos ^{4} \frac{x}{2} d x\)

Find the arc length of the curve over the given interval. $$ y=\ln x, \quad[1,5] $$

Find the arc length of the curve over the given interval. $$ y=\frac{1}{2} x^{2}, \quad[0,4] $$

State (if possible) the method or integration formula you would use to find the antiderivative. Explain why you chose that method or formula. Do not integrate. $$ \int \frac{e^{x}}{e^{x}+1} d x $$

Find the arc length of the graph of \(y=\sqrt{16-x^{2}}\) over the interval \([0,4]\).

Find any asymptotes and relative extrema that may exist and use a graphing utility to graph the function. (Hint: Some of the limits required in finding asymptotes have been found in preceding exercises.) $$ y=\frac{\ln x}{x} $$

State whether you would use integration by parts to evaluate the integral. If so, identify what you would use for \(u\) and \(d v .\) Explain your reasoning. $$ \int x \ln x d x $$

Use a computer algebra system to find the integral. Graph the antiderivatives for two different values of the constant of integration. \(\int \sin ^{2} x \cos ^{2} x d x\)