Exam 8: Integrals and Transcendental Functions

Verify the integration formula. - $\int \frac { \tanh ^ { - 1 } ( \ln x ) } { x } d x = \ln x \tanh ^ { - 1 } ( \ln x ) + \frac { 1 } { 2 } \ln \left( 1 - ( \ln x ) ^ { 2 } \right) + C$

Evaluate the integral. - $\int _ { \ln 2 } ^ { \ln 4 } \operatorname { coth } 10 x d x$

Find the derivative of y. - $y = \ln ( \operatorname { sech } ( 4 x + 5 ) )$

Evaluate the integral. - $\int _ { 1 } ^ { \mathrm { e } } 5 \mathrm { x } \ln 4 - 1 \mathrm { dx }$

Solve the differential equation. - $\frac { d y } { d x } = 3 \cos x \sec y$

Solve the problem. -A region in the first quadrant is bounded above by the curve $y = \cosh x$ , below by the curve $y = \sinh x$ , on the left by the $y$ -axis, and on the right by the line $x = 12$ . Find the volume of the solid generated by revolving the region about the $x$ -axis.

Solve the problem. -Suppose that the amount of oil pumped from a well decreases at the continuous rate of 12% per year. When, to the nearest year, will the well's output fall to one-eighth of its present value?

Provide an appropriate response. -Suppose you are looking for an item in an ordered list one million items long. Which would be better, a sequential search or a binary search? Why?

Evaluate the integral. - $\int _ { 0 } ^ { 5 } ( \sqrt { 7 } + 1 ) \times \sqrt { 7 } \mathrm { dx }$

Evaluate the integral. - $\int \left( e ^ { x } + e ^ { - x } \right) ^ { 2 } d x$

Find the derivative of y with respect to the appropriate variable. - $y = \sinh ^ { - 1 } ( \cos x )$

Solve the problem. -Find the length of the curve $y = \frac { x ^ { 2 } } { 4 } - \frac { 1 } { 2 } \ln x , 2 \leq x \leq 4$ .

Evaluate the integral in terms of natural logarithms. - $\int _ { 3 } ^ { 18 } \frac { d x } { x \sqrt { x ^ { 2 } + 9 } }$

Evaluate the integral. - $\int \frac { 2 e ^ { ( 2 \sin 6 x ) } } { \sec 6 x } d x$

Evaluate the integral in terms of natural logarithms. - $\int _ { 0 } ^ { \sqrt { 2 } / 2 } \frac { 32 \mathrm { dx } } { \sqrt { 1 + 16 \mathrm { x } ^ { 2 } } }$

Solve the problem. -A region in the first quadrant is bounded above by the curve $y = \tanh x$ , below by the $x$ -axis, on the left by the $y$ -axis, and on the right by the line $x = \ln 5$ . Find the volume of the solid generated by revolving the region about the $x$ -axis.

Solve the initial value problem. - $\frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dt } ^ { 2 } } = 4 - \mathrm { e } ^ { - \mathrm { t } } , \mathrm { y } ( 1 ) = \frac { - 1 } { \mathrm { e } } , \mathrm { y } ^ { \prime } ( 0 ) = - 6$

Evaluate the integral. - $\int \sinh 8 x d x$

Evaluate the integral in terms of natural logarithms. - $\int _ { 0 } ^ { 4 \sqrt { 3 } } \frac { \mathrm { dx } } { \sqrt { 16 + \mathrm { x } ^ { 2 } } }$

Find the derivative of y with respect to the appropriate variable. - $y = \operatorname { csch } ^ { - 1 } \left( \frac { 1 } { 3 } \right) ^ { \theta }$