91Ó°ÊÓ

In Exercises \(43-60,\) (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L'Hopital's Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). $$ \lim _{x \rightarrow 0^{+}} x^{1 / x} $$

Verify the integration formula. \(\int \frac{u^{2}}{(a+b u)^{2}} d u=\frac{1}{b^{3}}\left(b u-\frac{a^{2}}{a+b u}-2 a \ln |a+b u|\right)+C\)

Evaluating an Improper Integral In Exercises \(33-48\) determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{0}^{\infty} \frac{4}{\sqrt{x}(x+6)} d x $$

Trigonometric Substitution State the substitution you would make if you used trigonometric substitution for an integral involving the given radical, where \(a>0\) . Explain your reasoning. (a) \(\sqrt{a^{2}-u^{2}}\) (b) \(\sqrt{a^{2}+u^{2}}\) (c) \(\sqrt{u^{2}-a^{2}}\)

Volume and Centroid Consider the region bounded by the graphs of \(y=2 x /\left(x^{2}+1\right), y=0, x=0,\) and \(x=3 .\) Find the volume of the solid generated by revolving the region about the \(x\) -axis. Find the centroid of the region.

Finding an Indefinite Integral In Exercises \(47-56\) , find the indefinite integral. Use a computer algebra system to confirm your result. $$ \int \cot ^{3} 2 x d x $$

Finding an Indefinite Integral In Exercises \(47-56\) , find the indefinite integral. Use a computer algebra system to confirm your result. $$ \int \tan ^{5} \frac{x}{4} \sec ^{4} \frac{x}{4} d x $$

In Exercises \(43-60,\) (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L'Hopital's Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). $$ \lim _{x \rightarrow 0^{+}}\left(e^{x}+x\right)^{2 / x} $$

In Exercises 39–48, evaluate the definite integral. Use a graphing utility to confirm your result. $$ \int_{0}^{\pi / 8} x \sec ^{2} 2 x d x $$

Evaluating an Improper Integral In Exercises \(33-48\) determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{1}^{\infty} \frac{1}{x \ln x} d x $$