Problem 1

In Exercises \(1-6,\) identify \(u\) and \(d v\) for finding the integral using integration by parts. (Do not evaluate the integral.) $$\int x e^{2 x} d x$$

Problem 1

Select the correct anti derivative. $$\frac{d y}{d x}=\frac{x}{\sqrt{x^{2}+1}}$$ $$\begin{array}{ll}{\text { (a) } 2 \sqrt{x^{2}+1}+C} & {\text { (b) } \sqrt{x^{2}+1}+C} \\ {\text { (c) } \frac{1}{2} \sqrt{x^{2}+1}+C} & {\text { (d) } \ln \left(x^{2}+1\right)+C}\end{array}$$

Problem 1

In Exercises \(1-4,\) state the trigonometric substitution you would use to find the integral. Do not integrate. \(\int\left(9+x^{2}\right)^{-2} d x\)

Problem 1

In Exercises \(1-8,\) decide whether the integral is improper. Explain your reasoning. \(\int_{0}^{1} \frac{d x}{5 x-3}\)

Problem 1

In Exercises 1 and \(2,\) use a table of integrals with forms involving \(a+b u\) to find the integral. $$ \int \frac{x^{2}}{5+x} d x $$

Problem 1

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. \(\frac{4}{x^{2}-8 x}\)

Problem 1

Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to support your result. $$\lim _{x \rightarrow 0} \frac{\sin 4 x}{\sin 3 x}$$ $$\begin{array}{|c|c|c|c|c|c|}\hline x & {-0.1} & {-0.01} & {-0.001} & {0.001} & {0.01} & {0.1} \\ \hline f(x) & {} & {} \\ \hline\end{array}$$

Problem 2

Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to support your result. $$\lim _{x \rightarrow 0} \frac{1-e^{x}}{x}$$ $$\begin{array}{|c|c|c|c|c|}\hline x & {-0.1} & {-0.01} & {-0.001} & {0.001} & {0.01} & {0.1} \\ \hline f(x) & {} & {} \\ \hline\end{array}$$

Problem 2

In Exercises \(1-4,\) state the trigonometric substitution you would use to find the integral. Do not integrate. \(\int \sqrt{4-x^{2}} d x\)

Problem 2

In Exercises \(1-8,\) decide whether the integral is improper. Explain your reasoning. \(\int_{1}^{2} \frac{d x}{x^{3}}\)