Chapter 7

Evaluate the integrals. $$\int 4 \cosh (3 x-\ln 2) d x$$

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

Evaluate the integrals in Exercises \(29-50.\) $$\int_{\pi / 4}^{\pi / 2}\left(1+e^{\cot \theta}\right) \csc ^{2} \theta d \theta$$

Evaluate the integrals. $$\int \frac{d x}{\sqrt{1-4 x^{2}}}$$

Suppose that the differentiable function \(y=g(x)\) has an inverse and that the graph of \(g\) passes through the origin with slope 2 Find the slope of the graph of \(g^{-1}\) at the origin.

Evaluate the integrals. $$\int_{2}^{4} \frac{d x}{x \ln x}$$

Evaluate the integrals. $$\int \tanh \frac{x}{7} d x$$

Use l'Hôpital's rule to find the limits. $$\lim _{t \rightarrow \infty} \frac{e^{t}+t^{2}}{e^{t}-t}$$

Evaluate the integrals in Exercises \(29-50.\) $$\int e^{\sec \pi t} \sec \pi t \tan \pi t d t$$

a. Find the inverse of the function \(f(x)=m x,\) where \(m\) is a constant different from zero. b. What can you conclude about the inverse of a function \(y=f(x)\) whose graph is a line through the origin with a nonzero slope \(m ?\)

Evaluate the integrals. $$\int \frac{d x}{17+x^{2}}$$

Evaluate the integrals. $$\int_{2}^{4} \frac{d x}{x(\ln x)^{2}}$$

Evaluate the integrals. $$\int \operatorname{coth} \frac{\theta}{\sqrt{3}} d \theta$$

Use l'Hôpital's rule to find the limits. $$\lim _{x \rightarrow \infty} x^{2} e^{-x}$$

Evaluate the integrals in Exercises \(29-50.\) $$\int e^{\csc (\pi+t)} \csc (\pi+t) \cot (\pi+t) d t$$

Show that the graph of the inverse of \(f(x)=m x+b,\) where \(m\) and \(b\) are constants and \(m \neq 0,\) is a line with slope \(1 / m\) and \(y\) -intercept \(-b / m\).

Evaluate the integrals. $$\int \frac{d x}{9+3 x^{2}}$$

Evaluate the integrals. $$\int_{2}^{16} \frac{d x}{2 x \sqrt{\ln x}}$$

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

Use l'Hôpital's rule to find the limits. $$\lim _{x \rightarrow 0} \frac{x-\sin x}{x \tan x}$$

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

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 \(y=x ?\)

Evaluate the integrals. $$\int \frac{d x}{x \sqrt{25 x^{2}-2}}$$

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

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

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

Evaluate the integrals. $$\int \frac{d x}{x \sqrt{5 x^{2}-4}}$$

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

Evaluate the integrals. $$\int \frac{\operatorname{sech} \sqrt{t} \tanh \sqrt{t} d t}{\sqrt{t}}$$

Use l'Hôpital's rule to find the limits. $$\lim _{\theta \rightarrow 0} \frac{\theta-\sin \theta \cos \theta}{\tan \theta-\theta}$$

Evaluate the integrals in Exercises \(29-50.\) $$\int \frac{e^{r}}{1+e^{r}} d r$$

Show that increasing functions and decreasing functions are oneto-one. That is, show that for any \(x_{1}\) and \(x_{2}\) in \(I, x_{2} \neq x_{1}\) implies \(f\left(x_{2}\right) \neq f\left(x_{1}\right)\).

Evaluate the integrals. $$\int_{0}^{1} \frac{4 d s}{\sqrt{4-s^{2}}}$$

The frozen remains of a young Incan woman were discovered by archeologist Johan Reinhard on Mt. Ampato in Peru during an expedition in 1995 . a. How much of the original carbon-14 was present if the estimated age of the "Ice Maiden" was 500 years? b. If a \(1 \%\) error can occur in the carbon-14 measurement, what is the oldest possible age for the Ice Maiden?

Evaluate the integrals. $$\int \frac{\operatorname{csch}(\ln t) \operatorname{coth}(\ln t) d t}{t}$$

Use l'Hôpital's rule to find the limits. $$\lim _{x \rightarrow 0} \frac{\sin 3 x-3 x+x^{2}}{\sin x \sin 2 x}$$

Evaluate the integrals in Exercises \(29-50.\) $$\int \frac{d x}{1+e^{x}}$$

Evaluate the integrals. $$\int_{0}^{3 \sqrt{2} / 4} \frac{d s}{\sqrt{9-4 s^{2}}}$$

Evaluate the integrals. $$\int_{-1 / 4}^{\pi / 2} \cot t d t$$

Evaluate the integrals. $$\int_{\ln 2}^{\ln 4} \operatorname{coth} x d x$$

Find the limits $$\lim _{x \rightarrow 1^{+}} x^{1 /(1-x)}$$

Evaluate the integrals. $$\int_{0}^{2} \frac{d t}{8+2 t^{2}}$$

Evaluate the integrals. $$\int \frac{d x}{2 \sqrt{x}+2 x}$$

Evaluate the integrals. $$\int_{0}^{\ln 2} \tanh 2 x d x$$

Find the limits $$\lim _{x \rightarrow 1^{+}} x^{1 /(x-1)}$$

Solve the initial value problems in Exercises \(51-54.\) $$\frac{d y}{d t}=e^{-t} \sec ^{2}\left(\pi e^{-t}\right), \quad y(\ln 4)=2 / \pi$$

Evaluate the integrals. $$\int_{-2}^{2} \frac{d t}{4+3 t^{2}}$$