Chapter 5

In the following exercises, compute each integral using appropriate substitutions. $$\int \frac{\cos ^{-1}(2 t)}{\sqrt{1-4 t^{2}}} d t$$

In the following exercises, compute each integral using appropriate substitutions. $$\int \frac{e^{t} \cos ^{-1}\left(e^{t}\right)}{\sqrt{1-e^{2 t}}} d t$$

In the following exercises, compute each definite integral. $$\int_{0}^{1 / 2} \frac{\tan \left(\sin ^{-1} t\right)}{\sqrt{1-t^{2}}} d t$$

In the following exercises, compute each definite integral. $$\int_{1 / 4}^{1 / 2} \frac{\tan \left(\cos ^{-1} t\right)}{\sqrt{1-t^{2}}} d t$$

In the following exercises, compute each definite integral. $$\int_{0}^{1 / 2} \frac{\sin \left(\tan ^{-1} t\right)}{1+t^{2}} d t$$

In the following exercises, compute each definite integral. $$\int_{0}^{1 / 2} \frac{\cos \left(\tan ^{-1} t\right)}{1+t^{2}} d t$$

For \(\quad A>0, \quad\) compute \(\quad I(A)=\int_{-A}^{A} \frac{d t}{1+t^{2}} \quad\) and evaluate \(\lim _{a \rightarrow \infty} I(A),\) the area under the graph of \(\frac{1}{1+t^{2}}\) on \([-\infty, \infty]\).

For \(A>0, \quad\) compute \(I(A)=\int_{-A}^{A} \frac{d t}{1+t^{2}}\) and evaluate \(\lim _{a \rightarrow \infty} I(A),\) the area under the graph of \(\frac{1}{1+t^{2}}\) on \([-\infty, \infty]\)

For \(1 < B < \infty,\) compute \(I(B)=\int_{1}^{B} \frac{d t}{t \sqrt{t^{2}-1}}\) and evaluate \(\lim _{B \rightarrow \infty} I(B), \quad\) the area under the graph of \(\frac{1}{t \sqrt{t^{2}-1}}\) over \([1, \infty)\).

For \(1

Use the substitution \(u=\sqrt{2} \cot x\) and the identity \(1+\cot ^{2} x=\csc ^{2} x\) to evaluate \(\int \frac{d x}{1+\cos ^{2} x} .\) (Hint: Multiply the top and bottom of the integrand by \(\csc ^{2} x .\) )

[T] Approximate the points at which the graphs of \(f(x)=2 x^{2}-1\) and \(g(x)=\left(1+4 x^{2}\right)^{-3 / 2}\) intersect, and approximate the area between their graphs accurate to three decimal places.

True or False. Justify your answer with a proof or a counterexample. Assume all functions \(f\) and \(g\) are continuous over their domains. $$\int_{a}^{b} f(x)^{2} d x=\int_{a}^{b} f(x) d x \int_{a}^{b} f(x) d x$$

True or False. Justify your answer with a proof or a counterexample. Assume all functions \(f\) and \(g\) are continuous over their domains. If \(f(x) \leq g(x)\) for all \(x \in[a, b], \quad\) then \(\int_{a}^{b} f(x) \leq \int_{a}^{b} g(x)\)

True or False. Justify your answer with a proof or a counterexample. Assume all functions \(f\) and \(g\) are continuous over their domains. All continuous functions have an antiderivative.

Evaluate the Riemann sums \(L_{4}\) and \(R_{4}\) for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer. $$y=3 x^{2}-2 x+1 \text { over }[-1,1]$$

Evaluate the Riemann sums \(L_{4}\) and \(R_{4}\) for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer. $$y=\ln \left(x^{2}+1\right) \text { over }[0, e]$$

Evaluate the Riemann sums \(L_{4}\) and \(R_{4}\) for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer. $$y=x^{2} \sin x \text { over }[0, \pi]$$

Evaluate the Riemann sums \(L_{4}\) and \(R_{4}\) for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer. $$y=\sqrt{x}+\frac{1}{x} \text { over }[1,4]$$

Evaluate the following integrals. $$\int_{-1}^{1}\left(x^{3}-2 x^{2}+4 x\right) d x$$

Evaluate the following integrals. $$\int_{0}^{4} \frac{3 t}{\sqrt{1+6 t^{2}}} d t$$

Evaluate the following integrals. $$\int_{\pi / 3}^{\pi / 2} 2 \sec (2 \theta) \tan (2 \theta) d \theta$$

Find the antiderivative. $$\int \frac{d x}{(x+4)^{3}}$$

Find the antiderivative. $$\int x \ln \left(x^{2}\right) d x$$

Find the antiderivative. $$\int \frac{4 x^{2}}{\sqrt{1-x^{6}}} d x$$

Find the antiderivative. $$\int \frac{e^{2 x}}{1+e^{4 x}} d x$$

Find the derivative. $$\frac{d}{d t} \int_{0}^{t} \frac{\sin x}{\sqrt{1+x^{2}}} d x$$

Find the derivative. $$\frac{d}{d x} \int_{1}^{x^{3}} \sqrt{4-t^{2}} d t$$

Find the derivative. $$\frac{d}{d x} \int_{1}^{\ln (x)}\left(4 t+e^{t}\right) d t$$

Find the derivative. $$\frac{d}{d x} \int_{0}^{\cos x} e^{t^{2}} d t$$

The following problems consider the historic average cost per gigabyte of RAM on a computer. $$\begin{array}{|c|c|c|}\hline \text { Year } & {5 \text { -Year Change (s) }} \\\ \hline 1980 & {0} \\ \hline 1985 & {-5,468,750} \\ \hline 1990 & {-755,495} \\ \hline 1995 & {-73,005} \\ \hline 2000 & {-29,768} \\ \hline 2005 & {-918} \\ \hline 2010 & {-177} \\ \hline\end{array}$$ If the average cost per gigabyte of RAM in 2010 is \(\$ 12,\) find the average cost per gigabyte of \(\mathrm{RAM}\) in 1980 .

The following problems consider the historic average cost per gigabyte of RAM on a computer. $$\begin{array}{|c|c|c|}\hline \text { Year } & {5 \text { -Year Change (s) }} \\\ \hline 1980 & {0} \\ \hline 1985 & {-5,468,750} \\ \hline 1990 & {-755,495} \\ \hline 1995 & {-73,005} \\ \hline 2000 & {-29,768} \\ \hline 2005 & {-918} \\ \hline 2010 & {-177} \\ \hline\end{array}$$ The average cost per gigabyte of RAM can be approximated \(C(t)=8,500,000(0.65)^{t}\), where \(t\) is measured in years since \(1980,\) and \(C\) is cost in USS. Find the average cost per gigabyte of RAM for 1980 to 2010 .

The average cost per gigabyte of RAM can be approximated \(\quad\) by the function \(C(t)=8,500,000(0.65)^{t},\) where \(t\) is measured in years since 1980 , and \(C\) is cost in US\$. Find the average cost per gigabyte of RAM for 1980 to 2010 .

The following problems consider the historic average cost per gigabyte of RAM on a computer. $$\begin{array}{|c|c|c|}\hline \text { Year } & {5 \text { -Year Change (s) }} \\\ \hline 1980 & {0} \\ \hline 1985 & {-5,468,750} \\ \hline 1990 & {-755,495} \\ \hline 1995 & {-73,005} \\ \hline 2000 & {-29,768} \\ \hline 2005 & {-918} \\ \hline 2010 & {-177} \\ \hline\end{array}$$ Find the average cost of 1 \(\mathrm{GB}\) RAM for 2005 to 2010 .

The following problems consider the historic average cost per gigabyte of RAM on a computer. $$\begin{array}{|c|c|c|}\hline \text { Year } & {5 \text { -Year Change (s) }} \\\ \hline 1980 & {0} \\ \hline 1985 & {-5,468,750} \\ \hline 1990 & {-755,495} \\ \hline 1995 & {-73,005} \\ \hline 2000 & {-29,768} \\ \hline 2005 & {-918} \\ \hline 2010 & {-177} \\ \hline\end{array}$$ The velocity of a bullet from a rifle can be approximated by \(v(t)=6400 t^{2}-6505 t+2686,\) where \(t\) is seconds after the shot and \(v\) is the velocity measured in feet per second. This equation only models the velocity for the first half-second after the shot: \(0 \leq t \leq 0.5\) . What is the total distance the bullet travels in 0.5 \(\sec ?\)

The velocity of a bullet from a rifle can be approximated by \(v(t)=6400 t^{2}-6505 t+2686,\) where \(t\) is seconds after the shot and \(v\) is the velocity measured in feet per second. This equation only models the velocity for the first half-second after the shot: \(0 \leq t \leq 0.5\). What is the total distance the bullet travels in 0.5 sec?