91Ó°ÊÓ

Evaluate the integrals. $$\int \operatorname{sech}^{2}\left(x-\frac{1}{2}\right) d x$$

Evaluate the integrals in Exercises \(29-50.\) $$\int_{\ln (\pi / 6)}^{\ln (\pi / 2)} 2 e^{v} \cos e^{v} d v$$

Evaluate the integrals. $$\int \frac{3 \sec ^{2} t}{6+3 \tan t} d t$$

a. Find the inverse of \(f(x)=x+1 .\) Graph \(f\) and its inverse together. Add the line \(y=x\) to your sketch, drawing it with dashes or dots for contrast. b. Find the inverse of \(f(x)=x+b(b\) constant). How is the graph of \(f^{-1}\) related to the graph of \(f ?\) c. What can you conclude about the inverses of functions whose graphs are lines parallel to the line \(y=x ?\)

Use l'Hôpital's rule to find the limits. $$\lim _{x \rightarrow 0} \frac{\left(e^{x}-1\right)^{2}}{x \sin x}$$

a. Find the inverse of \(f(x)=-x+1 .\) Graph the line \(y=-x+1\) together with the line \(y=x .\) At what angle do the lines intersect? b. Find the inverse of \(f(x)=-x+b(b \text { constant }) .\) What angle does the line \(y=-x+b\) make with the line \(y=x ?\) c. What can you conclude about the inverses of functions whose graphs are lines perpendicular to the line \(y=x ?\)

Evaluate the integrals in Exercises \(29-50.\) $$\int_{0}^{\sqrt{\ln \pi}} 2 x e^{x^{2}} \cos \left(e^{x^{2}}\right) d x$$

\(A\) painting attributed to Vermeer \((1632-1675)\) which should contain no more than \(96.2 \%\) of its original carbon-14, contains 99.5\% instead. About how old is the forgery?

Evaluate the integrals. $$\int \operatorname{csch}^{2}(5-x) d x$$

Evaluate the integrals. $$\int \frac{\sec y \tan y}{2+\sec y} d y$$