91Ó°ÊÓ

In Exercises \(35-64\) , use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. $$ \int_{\pi}^{\infty} \frac{2+\cos x}{x} d x $$

Use reduction formulas to evaluate the integrals. \(\int 8 \cos ^{4} 2 \pi t d t\)

Find, to two decimal places, the areas of the surfaces generated by revolving the curves in Exercises \(53-56\) about the \(x\) -axis. \(y=x+\sin 2 x, \quad-2 \pi / 3 \leq x \leq 2 \pi / 3 \quad\) (the curve in Section \(4.4,\) Exercise 5\()\)

In Exercises \(35-64\) , use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. $$ \int_{\pi}^{\infty} \frac{1+\sin x}{x^{2}} d x $$

Use reduction formulas to evaluate the integrals. \(\int 3 \cos ^{5} 3 y d y\)

Find, to two decimal places, the areas of the surfaces generated by revolving the curves in Exercises \(53-56\) about the \(x\) -axis. \(y=\frac{x}{12} \sqrt{36-x^{2}}, 0 \leq x \leq 6\) (the surface of the plumb bob in Section \(6.1,\) Exercise 56\()\)

Use reduction formulas to evaluate the integrals. \(\int \sin ^{2} 2 \theta \cos ^{3} 2 \theta d \theta\)

Use numerical integration to estimate the value of $$ \sin ^{-1} 0.6=\int_{0}^{0.6} \frac{d x}{\sqrt{1-x^{2}}} $$ For reference, \(\sin ^{-1} 0.6=0.64350\) to five decimal places.

Evaluate each integral in Exercises \(57-62\) by multiplying by a form of 1 and using a substitution (if necessary) to reduce it to standard form. $$ \int \frac{1}{1+\sin x} d x $$

In Exercises \(35-64\) , use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. $$ \int_{4}^{\infty} \frac{2 d t}{t^{3 / 2}-1} $$