91Ó°ÊÓ

Evaluate the integrals in Exercises 37-54. $$\int \frac{3 \sec ^{2} t}{6+3 \tan t} d t$$

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

Evaluate the integrals in Exercises \(41-60\) $$\int \operatorname{sech}^{2}\left(x-\frac{1}{2}\right) d x$$

Evaluate the integrals in Exercises 37-54. $$\int \frac{\sec y \tan y}{2+\sec y} d y$$

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

Evaluate the integrals in Exercises \(43-66\) $$ \int \frac{d x}{x \sqrt{5 x^{2}-4}} $$

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

Use l'Hopital's rule to find the limits in Exercises \(7-50\) $$\lim _{x \rightarrow 0} \frac{\left(e^{x}-1\right)^{2}}{x \sin x}$$

Evaluate the integrals in Exercises \(43-66\) $$ \int_{0}^{1} \frac{4 d s}{\sqrt{4-s^{2}}} $$

Evaluate the integrals. $$\int \frac{e^{r}}{1+e^{r}} d r$$