Question 1

Evaluate the integral in terms of natural logarithms.
-$$\int _ { 6 / 5 } ^ { 7 / 2 } \frac { d x } { 1 - x ^ { 2 } }$$

Accepted Answer

A)  $\ln 2$ 
B)  $\frac { 1 } { 2 } \ln \left( \frac { 9 } { 55 } \right)$ 
C)  $\ln \left( \frac { 7 } { 11 } \right)$ 
D)  $\frac { 1 } { 2 } \ln \left( - \frac { 9 } { 55 } \right)$ 
A)  $\ln 2$ 
B)  $\frac { 1 } { 2 } \ln \left( \frac { 9 } { 55 } \right)$ 
C)  $\ln \left( \frac { 7 } { 11 } \right)$ 
D)  $\frac { 1 } { 2 } \ln \left( - \frac { 9 } { 55 } \right)$

Question 2

Solve the problem.
-Find the half-life of the radioactive element radium, assuming that its decay constant is $\mathrm { k } = 4.332 \times 10 ^ { - 4 }$, with time measured in years.

Accepted Answer

A)  800 years 
B)  1400 years 
C)  2308 years 
D)  1600 years 
A)  800 years 
B)  1400 years 
C)  2308 years 
D)  1600 years

Question 3

Evaluate the integral.
-$$\int _ { 1 } ^ { 9 } 8 \frac { \sinh \sqrt { x } } { \sqrt { x } } d x$$

Accepted Answer

A)  $8 \left( \mathrm { e } ^ { 3 } - \mathrm { e } \right)$ 
B)  $8 \left( \mathrm { e } ^ { 3 } + \mathrm { e } ^ { - 3 } - \mathrm { e } - \frac { 1 } { \mathrm { e } } \right)$ 
C)  $16 \left( \mathrm { e } ^ { 3 } + \mathrm { e } ^ { - 3 } - \mathrm { e } - \frac { 1 } { \mathrm { e } } \right)$ 
D)  $- 8$ 
A)  $8 \left( \mathrm { e } ^ { 3 } - \mathrm { e } \right)$ 
B)  $8 \left( \mathrm { e } ^ { 3 } + \mathrm { e } ^ { - 3 } - \mathrm { e } - \frac { 1 } { \mathrm { e } } \right)$ 
C)  $16 \left( \mathrm { e } ^ { 3 } + \mathrm { e } ^ { - 3 } - \mathrm { e } - \frac { 1 } { \mathrm { e } } \right)$ 
D)  $- 8$

Question 4

Evaluate the integral.
-$\int \operatorname { csch } ^ { 2 } \left( 3 - \frac { x } { 8 } \right) d x$

Accepted Answer

A)  $- \operatorname { coth } \left( 3 - \frac { x } { 8 } \right) + C$ 
B)  $8 \tanh \left( 3 - \frac { x } { 8 } \right) + C$ 
C)  $\frac { 8 } { 3 } \operatorname { csch } ^ { 3 } \left( 3 - \frac { x } { 8 } \right) + C$ 
D)  $8 \operatorname { coth } \left( 3 - \frac { x } { 8 } \right) + C$ 
A)  $- \operatorname { coth } \left( 3 - \frac { x } { 8 } \right) + C$ 
B)  $8 \tanh \left( 3 - \frac { x } { 8 } \right) + C$ 
C)  $\frac { 8 } { 3 } \operatorname { csch } ^ { 3 } \left( 3 - \frac { x } { 8 } \right) + C$ 
D)  $8 \operatorname { coth } \left( 3 - \frac { x } { 8 } \right) + C$

Question 5

Provide an appropriate response.
-

Accepted Answer

The answer of Provide an appropriate response.
- ...

Question 6

Determine if the given function y = f(x) is a solution of the accompanying differential equation.
-Differential equation: $9 x y ^ { \prime } + 9 y = \cos x$ Initial condition: $y ( \pi ) = 0$
Solution candidate: $y = \frac { \sin x } { 9 x }$

Accepted Answer

A) True 
 B)False

Question 7

Evaluate the integral.
-$$\int _ { \ln 2 } ^ { \ln 5 } \tanh x d x$$

Accepted Answer

A)  $\ln \frac { 52 } { 25 }$ 
B)  $\ln 2$ 
C)  $\frac { 27 } { 10 }$ 
D)  $\ln \frac { 27 } { 10 }$ 
A)  $\ln \frac { 52 } { 25 }$ 
B)  $\ln 2$ 
C)  $\frac { 27 } { 10 }$ 
D)  $\ln \frac { 27 } { 10 }$

Question 8

Verify the integration formula.
-$$\int 8 \tanh 8 x d x = \ln \cosh ^ { - 1 } 8 x + C$$

Accepted Answer

A) True 
 B)False

Question 9

Find the derivative of y.
-$$y = \cosh x ^ { 5 }$$

Accepted Answer

A)  $\sinh x ^ { 5 }$ 
B)  $- 5 x ^ { 4 } \sinh x ^ { 5 }$ 
C)  $5 x ^ { 4 } \sinh x ^ { 5 }$ 
D)  $- \sinh x ^ { 5 }$ 
A)  $\sinh x ^ { 5 }$ 
B)  $- 5 x ^ { 4 } \sinh x ^ { 5 }$ 
C)  $5 x ^ { 4 } \sinh x ^ { 5 }$ 
D)  $- \sinh x ^ { 5 }$

Question 10

A value of $\sinh x$ or $\cosh x$ is given. Use the definitions and the identity $\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$ to find the value of the other indicated hyperbolic function.
-$\sinh x = \frac { 5 } { 12 } , \cosh x =$

Accepted Answer

A)  $\frac { 12 } { 13 }$ 
B)  $\frac { 169 } { 144 }$ 
C)  $\frac { 13 } { 12 }$ 
D)  $- \frac { 13 } { 12 }$ 
A)  $\frac { 12 } { 13 }$ 
B)  $\frac { 169 } { 144 }$ 
C)  $\frac { 13 } { 12 }$ 
D)  $- \frac { 13 } { 12 }$

Question 11

Evaluate the integral.
-$\int 5 x \sqrt { 3 } + 3 d x$

Accepted Answer

A)  $\frac { 5 x \sqrt { 3 } + 3 } { \ln x } + C$ 
B)  $\frac { 5 x \sqrt { 3 } + 2 } { \sqrt { 3 } + 2 } + C$ 
C)  $\frac { 5 x \sqrt { 3 } + 4 } { \sqrt { 3 } + 4 } + C$ 
D)  $\frac { 5 } { \sqrt { 3 } + 4 } + C$ 
A)  $\frac { 5 x \sqrt { 3 } + 3 } { \ln x } + C$ 
B)  $\frac { 5 x \sqrt { 3 } + 2 } { \sqrt { 3 } + 2 } + C$ 
C)  $\frac { 5 x \sqrt { 3 } + 4 } { \sqrt { 3 } + 4 } + C$ 
D)  $\frac { 5 } { \sqrt { 3 } + 4 } + C$

Question 12

Solve the differential equation.
-$$\frac { d y } { d x } = e ^ { 5 x - 5 y }$$

Accepted Answer

A)  $y = 5 e ^ { 5 x } + C$ 
B)  $y = 5 \ln \left( e ^ { 5 x } + C \right)$ 
C)  $y = \ln \left( e ^ { 5 x } + C \right)$ 
D)  $y = \frac { 1 } { 5 } \ln \left( e ^ { 5 x } + C \right)$ 
A)  $y = 5 e ^ { 5 x } + C$ 
B)  $y = 5 \ln \left( e ^ { 5 x } + C \right)$ 
C)  $y = \ln \left( e ^ { 5 x } + C \right)$ 
D)  $y = \frac { 1 } { 5 } \ln \left( e ^ { 5 x } + C \right)$

Question 13

Solve the problem.
-The intensity $\mathrm { L } ( \mathrm { x } )$ of light $x \mathrm { ft }$ beneath the surface of a lake satisfies the differential equation $\frac { \mathrm { dL } } { \mathrm { dx } } = - 0.07 \mathrm {~L} .$ At what depth, to the nearest foot, is the intensity one tenth the intensity at the surface?

Accepted Answer

A)  22 ft 
B)  49 ft 
C)  16 ft 
D)  33 ft 
A)  22 ft 
B)  49 ft 
C)  16 ft 
D)  33 ft

Question 14

Find the derivative of y.
-$$y = - 4 t ^ { 3 } \tanh \left( \frac { 1 } { t ^ { 2 } } \right)$$

Accepted Answer

A)  $- 12 t ^ { 2 } \tanh \left( \frac { 1 } { t ^ { 2 } } \right) + 8 \operatorname { sech } ^ { 2 } \left( \frac { 1 } { t ^ { 2 } } \right)$ 
B)  $- 12 t ^ { 2 } \tanh \left( \frac { 1 } { t ^ { 2 } } \right) + 8 \operatorname { sech } \left( \frac { 1 } { t ^ { 2 } } \right)$ 
C)  $- 12 \mathrm { t } ^ { 2 } \tanh \left( \frac { 1 } { \mathrm { t } ^ { 2 } } \right) + 4 \operatorname { sech } ^ { 2 } \left( \frac { 1 } { \mathrm { t } ^ { 2 } } \right)$ 
D)  $- 12 t ^ { 2 } \tanh \left( \frac { 1 } { t ^ { 2 } } \right) - 8 \operatorname { sech } ^ { 2 } \left( \frac { 1 } { t ^ { 2 } } \right)$ 
A)  $- 12 t ^ { 2 } \tanh \left( \frac { 1 } { t ^ { 2 } } \right) + 8 \operatorname { sech } ^ { 2 } \left( \frac { 1 } { t ^ { 2 } } \right)$ 
B)  $- 12 t ^ { 2 } \tanh \left( \frac { 1 } { t ^ { 2 } } \right) + 8 \operatorname { sech } \left( \frac { 1 } { t ^ { 2 } } \right)$ 
C)  $- 12 \mathrm { t } ^ { 2 } \tanh \left( \frac { 1 } { \mathrm { t } ^ { 2 } } \right) + 4 \operatorname { sech } ^ { 2 } \left( \frac { 1 } { \mathrm { t } ^ { 2 } } \right)$ 
D)  $- 12 t ^ { 2 } \tanh \left( \frac { 1 } { t ^ { 2 } } \right) - 8 \operatorname { sech } ^ { 2 } \left( \frac { 1 } { t ^ { 2 } } \right)$

Question 15

Solve the problem.
-Find the volume of the solid that is generated by revolving the area bounded by $y = \frac { 7 } { \sqrt { 6 x + 1 } } , x = 0 , x = 3$, and $y = 0$ about the $x$-axis.

Accepted Answer

A)  $\frac { 7 } { 6 } \pi \ln ( 6 )$ 
B)  $\frac { 49 } { 6 } \pi \ln ( 6 )$ 
C)  $\frac { 7 \sqrt { 6 } } { 6 } \pi \ln ( 19 )$ 
D)  $\frac { 49 } { 6 } \pi \ln ( 19 )$ 
A)  $\frac { 7 } { 6 } \pi \ln ( 6 )$ 
B)  $\frac { 49 } { 6 } \pi \ln ( 6 )$ 
C)  $\frac { 7 \sqrt { 6 } } { 6 } \pi \ln ( 19 )$ 
D)  $\frac { 49 } { 6 } \pi \ln ( 19 )$

Question 16

Evaluate the integral.
-$$\int _ { 0 } ^ { \sqrt { \ln \pi } } 2 x e ^ { x ^ { 2 } } \sin \left( e ^ { x ^ { 2 } } \right) d x$$

Accepted Answer

A)  1 + cos 1 
B)  -1 
C)  1 
D)  1 - cos 1 
A)  1 + cos 1 
B)  -1 
C)  1 
D)  1 - cos 1

Question 17

Find the derivative of y.
-$$y = \sinh ^ { 2 } 4 x$$

Accepted Answer

A)  2 sinh 4x cosh 4x 
B)  2 cosh 4x 
C)  8 cosh 4x 
D)  8 sinh 4x cosh 4x 
A)  2 sinh 4x cosh 4x 
B)  2 cosh 4x 
C)  8 cosh 4x 
D)  8 sinh 4x cosh 4x

Question 18

Evaluate the integral in terms of natural logarithms.
-$$\int _ { 0 } ^ { 11 \pi } \frac { - \sin x d x } { \sqrt { 1 + \cos ^ { 2 } x } }$$

Accepted Answer

A)  $\ln 2$ 
B)  $\ln \left( \frac { - 1 + \sqrt { 2 } } { 1 + \sqrt { 2 } } \right)$ 
C)  $- 2$ 
D)  0 
A)  $\ln 2$ 
B)  $\ln \left( \frac { - 1 + \sqrt { 2 } } { 1 + \sqrt { 2 } } \right)$ 
C)  $- 2$ 
D)  0

Question 19

Find the derivative of y with respect to the appropriate variable.
-$$y = \sinh ^ { - 1 } \sqrt { 11 x }$$

Accepted Answer

A)  $\frac { 11 } { 2 \sqrt { 11 x ( 11 x - 1 ) } }$ 
B)  $\frac { 11 } { 2 \sqrt { 11 x ( 1 + 11 x ) } }$ 
C)  $\frac { 1 } { \sqrt { 1 + 11 x } }$ 
D)  $\frac { 1 } { 2 \sqrt { 11 x ( 1 + 11 x ) } }$ 
A)  $\frac { 11 } { 2 \sqrt { 11 x ( 11 x - 1 ) } }$ 
B)  $\frac { 11 } { 2 \sqrt { 11 x ( 1 + 11 x ) } }$ 
C)  $\frac { 1 } { \sqrt { 1 + 11 x } }$ 
D)  $\frac { 1 } { 2 \sqrt { 11 x ( 1 + 11 x ) } }$

Question 20

Evaluate the integral.
-$$\int _ { \pi / 16 } ^ { \pi / 8 } 4 \cot ( 4 \theta ) d \theta$$

Accepted Answer

A)  $\ln 2$ 
B)  $- \frac { \ln 2 } { 2 }$ 
C)  $\frac { \ln 2 } { 2 }$ 
D)  $\frac { \ln 8 } { 2 }$ 
A)  $\ln 2$ 
B)  $- \frac { \ln 2 } { 2 }$ 
C)  $\frac { \ln 2 } { 2 }$ 
D)  $\frac { \ln 8 } { 2 }$

Evaluate the integral in terms of natural logarithms. - $\int _ { 6 / 5 } ^ { 7 / 2 } \frac { d x } { 1 - x ^ { 2 } }$

Solve the problem. -Find the half-life of the radioactive element radium, assuming that its decay constant is $\mathrm { k } = 4.332 \times 10 ^ { - 4 }$ , with time measured in years.

Evaluate the integral. - $\int _ { 1 } ^ { 9 } 8 \frac { \sinh \sqrt { x } } { \sqrt { x } } d x$

Evaluate the integral. - $\int \operatorname { csch } ^ { 2 } \left( 3 - \frac { x } { 8 } \right) d x$

Provide an appropriate response. -

Determine if the given function y = f(x) is a solution of the accompanying differential equation. -Differential equation: $9 x y ^ { \prime } + 9 y = \cos x$ Initial condition: $y ( \pi ) = 0$ Solution candidate: $y = \frac { \sin x } { 9 x }$

Evaluate the integral. - $\int _ { \ln 2 } ^ { \ln 5 } \tanh x d x$

Verify the integration formula. - $\int 8 \tanh 8 x d x = \ln \cosh ^ { - 1 } 8 x + C$

Find the derivative of y. - $y = \cosh x ^ { 5 }$

A value of $\sinh x$ or $\cosh x$ is given. Use the definitions and the identity $\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$ to find the value of the other indicated hyperbolic function. - $\sinh x = \frac { 5 } { 12 } , \cosh x =$

Evaluate the integral. - $\int 5 x \sqrt { 3 } + 3 d x$

Solve the differential equation. - $\frac { d y } { d x } = e ^ { 5 x - 5 y }$

Find the derivative of y. - $y = - 4 t ^ { 3 } \tanh \left( \frac { 1 } { t ^ { 2 } } \right)$

Solve the problem. -Find the volume of the solid that is generated by revolving the area bounded by $y = \frac { 7 } { \sqrt { 6 x + 1 } } , x = 0 , x = 3$ , and $y = 0$ about the $x$ -axis.

Evaluate the integral. - $\int _ { 0 } ^ { \sqrt { \ln \pi } } 2 x e ^ { x ^ { 2 } } \sin \left( e ^ { x ^ { 2 } } \right) d x$

Find the derivative of y. - $y = \sinh ^ { 2 } 4 x$

Evaluate the integral in terms of natural logarithms. - $\int _ { 0 } ^ { 11 \pi } \frac { - \sin x d x } { \sqrt { 1 + \cos ^ { 2 } x } }$

Find the derivative of y with respect to the appropriate variable. - $y = \sinh ^ { - 1 } \sqrt { 11 x }$

Evaluate the integral. - $\int _ { \pi / 16 } ^ { \pi / 8 } 4 \cot ( 4 \theta ) d \theta$

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Exam 8: Integrals and Transcendental Functions

Evaluate the integral in terms of natural logarithms. - ∫6/57/2dx1−x2\int _ { 6 / 5 } ^ { 7 / 2 } \frac { d x } { 1 - x ^ { 2 } }∫6/57/2​1−x2dx​

Solve the problem. -Find the half-life of the radioactive element radium, assuming that its decay constant is k=4.332×10−4\mathrm { k } = 4.332 \times 10 ^ { - 4 }k=4.332×10−4 , with time measured in years.

Evaluate the integral. - ∫198sinh⁡xxdx\int _ { 1 } ^ { 9 } 8 \frac { \sinh \sqrt { x } } { \sqrt { x } } d x∫19​8x​sinhx​​dx

Evaluate the integral. - ∫csch⁡2(3−x8)dx\int \operatorname { csch } ^ { 2 } \left( 3 - \frac { x } { 8 } \right) d x∫csch2(3−8x​)dx