Question 1

Rewrite the expression in terms of exponentials and simplify the results.
-$\cosh 3 x + \sinh 3 x$

Accepted Answer

A)  $3 x$ 
B)  $e ^ { 3 x }$ 
C)  $\mathrm { e } ^ { 3 x } - \mathrm { e } ^ { - 3 \mathrm { x } }$ 
D)  $2 \mathrm { e } ^ { 3 \mathrm { x } }$ 
A)  $3 x$ 
B)  $e ^ { 3 x }$ 
C)  $\mathrm { e } ^ { 3 x } - \mathrm { e } ^ { - 3 \mathrm { x } }$ 
D)  $2 \mathrm { e } ^ { 3 \mathrm { x } }$

Question 2

Evaluate the integral.
-$$\int \frac { 21 \mathrm { e } \sqrt { 7 x } } { 2 \sqrt { x } } \mathrm { dx }$$

Accepted Answer

A)  $\sqrt { 7 } e ^{\sqrt { 7 x }} + C$ 
B)  $\frac { 21 } { 2 } e ^{\sqrt { 7 x }} + C$ 
C)  $21 \mathrm { e } ^{\sqrt { 7 x }} + C$ 
D)  $3 \sqrt { 7 } \mathrm { e }^{ \sqrt { 7 x }} + C$ 
A)  $\sqrt { 7 } e ^{\sqrt { 7 x }} + C$ 
B)  $\frac { 21 } { 2 } e ^{\sqrt { 7 x }} + C$ 
C)  $21 \mathrm { e } ^{\sqrt { 7 x }} + C$ 
D)  $3 \sqrt { 7 } \mathrm { e }^{ \sqrt { 7 x }} + C$

Question 3

Rewrite the expression in terms of exponentials and simplify the results.
-$( \sinh x + \cosh x ) ^ { 7 }$

Accepted Answer

A)  $e ^ { x ^ { 7 } }$ 
B)  $\mathrm { e } ^ { 7 \mathrm { x } }$ 
C)  $\mathrm { e } ^ { 7 x } - \mathrm { e } ^ { - 7 \mathrm { x } }$ 
D)  $\frac { e ^ { 7 x } } { 4 }$ 
A)  $e ^ { x ^ { 7 } }$ 
B)  $\mathrm { e } ^ { 7 \mathrm { x } }$ 
C)  $\mathrm { e } ^ { 7 x } - \mathrm { e } ^ { - 7 \mathrm { x } }$ 
D)  $\frac { e ^ { 7 x } } { 4 }$

Question 4

Evaluate the integral.
-$$\int \frac { d x } { x ( 4 + 6 \ln x ) }$$

Accepted Answer

A)  $\frac { 1 } { 6 } \ln | 4 + 6 \ln x | + C$ 
B)  $\frac { 2 } { 3 } \ln | 4 + 6 \ln x | + C$ 
C)  $\frac { 1 } { 4 } \ln | 4 + 6 \ln x | + C$ 
D)  $\frac { 1 } { 6 } \ln | 6 + 4 \ln x | + C$ 
A)  $\frac { 1 } { 6 } \ln | 4 + 6 \ln x | + C$ 
B)  $\frac { 2 } { 3 } \ln | 4 + 6 \ln x | + C$ 
C)  $\frac { 1 } { 4 } \ln | 4 + 6 \ln x | + C$ 
D)  $\frac { 1 } { 6 } \ln | 6 + 4 \ln x | + C$

Question 5

Evaluate the integral in terms of an inverse hyperbolic function.
-$$\int _ { 1 / 7 } ^ { 2 / 11 } \frac { d x } { x \sqrt { 25 - 64 x ^ { 2 } } }$$

Accepted Answer

A)  $- \frac { 1 } { 5 } \operatorname { sech } ^ { - 1 } \left( \frac { 16 } { 55 } \right) + \frac { 1 } { 5 } \operatorname { sech } ^ { - 1 } \left( \frac { 8 } { 35 } \right)$ 
B)  $\frac { 1 } { 5 } \tanh ^ { - 1 } \left( \frac { 20 } { 11 } \right) - \frac { 1 } { 5 } \tanh ^ { - 1 } \left( \frac { 5 } { 56 } \right)$ 
C)  $\frac { 1 } { 5 } \tanh ^ { - 1 } \left( \frac { 16 } { 55 } \right) - \frac { 1 } { 5 } \tanh ^ { - 1 } \left( \frac { 8 } { 35 } \right)$ 
D)  $- \frac { 1 } { 5 } \operatorname { sech } ^ { - 1 } \left( \frac { 5 } { 11 } \right) + \frac { 1 } { 5 } \operatorname { sech } ^ { - 1 } \left( \frac { 5 } { 56 } \right)$ 
A)  $- \frac { 1 } { 5 } \operatorname { sech } ^ { - 1 } \left( \frac { 16 } { 55 } \right) + \frac { 1 } { 5 } \operatorname { sech } ^ { - 1 } \left( \frac { 8 } { 35 } \right)$ 
B)  $\frac { 1 } { 5 } \tanh ^ { - 1 } \left( \frac { 20 } { 11 } \right) - \frac { 1 } { 5 } \tanh ^ { - 1 } \left( \frac { 5 } { 56 } \right)$ 
C)  $\frac { 1 } { 5 } \tanh ^ { - 1 } \left( \frac { 16 } { 55 } \right) - \frac { 1 } { 5 } \tanh ^ { - 1 } \left( \frac { 8 } { 35 } \right)$ 
D)  $- \frac { 1 } { 5 } \operatorname { sech } ^ { - 1 } \left( \frac { 5 } { 11 } \right) + \frac { 1 } { 5 } \operatorname { sech } ^ { - 1 } \left( \frac { 5 } { 56 } \right)$

Question 6

Evaluate the integral.
-$\int \frac { \operatorname { csch } ( \ln x ) \operatorname { coth } ( \ln x ) } { 4 x } d x$

Accepted Answer

A)  $\frac { 1 } { 4 } \operatorname { csch } ( \ln x ) + C$ 
B)  $4 \operatorname { sech } ( \ln x ) + C$ 
C)  $x \operatorname { csch } ( \ln x ) + C$ 
D)  $\frac { - 1 } { 4 } \operatorname { csch } ( \ln x ) + C$ 
A)  $\frac { 1 } { 4 } \operatorname { csch } ( \ln x ) + C$ 
B)  $4 \operatorname { sech } ( \ln x ) + C$ 
C)  $x \operatorname { csch } ( \ln x ) + C$ 
D)  $\frac { - 1 } { 4 } \operatorname { csch } ( \ln x ) + C$

Question 7

Verify the integration formula.
-$$\int 2 x ^ { 3 } \operatorname { sech } ^ { - 1 } x ^ { 2 } d x = \frac { x ^ { 4 } } { 2 } \operatorname { sech } ^ { - 1 } x + \frac { 1 } { 2 } \sqrt { 1 - x ^ { 4 } } + C$$

Accepted Answer

A) True 
 B)False

Question 8

Rewrite the expression in terms of exponentials and simplify the results.
-$\cosh 9 x - \sinh 9 x$

Accepted Answer

A)  $- 9 x$ 
B)  $e ^ { 9 x } - e ^ { - 9 x }$ 
C)  $\mathrm { e } ^ { - 9 \mathrm { x } }$ 
D)  $2 e ^ { - 9 x }$ 
A)  $- 9 x$ 
B)  $e ^ { 9 x } - e ^ { - 9 x }$ 
C)  $\mathrm { e } ^ { - 9 \mathrm { x } }$ 
D)  $2 e ^ { - 9 x }$

Question 9

Solve the problem.
-The velocity of a body of mass $m$ falling from rest under the action of gravity is given by the equation $\mathrm { v } = \sqrt { \frac { \mathrm { mg } } { \mathrm { k } } } \tanh \left( \sqrt { \frac { \mathrm { gk } } { \mathrm { m } } } \mathrm { t } \right)$, where $\mathrm { k }$ is a constant that depends on the body's aerodynamic properties and the density of the air, $g$ is the gravitational constant, and $t$ is the number of seconds into the fall. Find the limiting velocity, $\lim _ { \mathrm { t } \rightarrow } \mathrm { v }$, of a $420 \mathrm { lb }$. skydiver ( $\mathrm { mg } = 420$ ) when $\mathrm { k } = 0.006$.

Accepted Answer

A)  264.58 ft/sec 
B)  0.01 ft/sec 
C)  There is no limiting speed. 
D)  83.67 ft/sec 
A)  264.58 ft/sec 
B)  0.01 ft/sec 
C)  There is no limiting speed. 
D)  83.67 ft/sec

Question 10

Rewrite the expression in terms of exponentials and simplify the results.
-$\ln ( \cosh 5 x - \sinh 5 x ) + \ln ( \cosh 8 x + \sinh 8 x )$

Accepted Answer

A)  3 
B)  $\ln \left( e ^ { 8 x } - e ^ { - 5 x } \right)$ 
C)  $13 x$ 
D)  $3 x$ 
A)  3 
B)  $\ln \left( e ^ { 8 x } - e ^ { - 5 x } \right)$ 
C)  $13 x$ 
D)  $3 x$

Question 11

Solve the initial value problem.
-$$\frac { d y } { d x } = - 5 e ^ { - x } \sec e ^ { - x } \tan e ^ { - x } , y ( 0 ) = 5 \sec 1 + 6$$

Accepted Answer

A)  $y = 5 \sec e ^ { - x } + 6$ 
B)  $y = - 5 \sec x + 1$ 
C)  $y = - 5 \sec e ^ { - x + 1 }$ 
D)  $y = 5 \tan \mathrm { e } ^ { - \mathrm { x } } + 6$ 
A)  $y = 5 \sec e ^ { - x } + 6$ 
B)  $y = - 5 \sec x + 1$ 
C)  $y = - 5 \sec e ^ { - x + 1 }$ 
D)  $y = 5 \tan \mathrm { e } ^ { - \mathrm { x } } + 6$

Question 12

Verify the integration formula.
-$$\int \operatorname { coth } ^ { - 1 } \sqrt { x } d x = ( x - 1 ) \operatorname { coth } ^ { - 1 } \sqrt { x } + \sqrt { x } + C$$

Accepted Answer

A) True 
 B)False

Question 13

Express the value of the inverse hyperbolic function in terms of natural logarithms.
-$\cosh ^ { - 1 } 7$

Accepted Answer

A)  $\ln ( 7 + \sqrt { 48 } )$ 
B)  $\ln ( 7 + \sqrt { 50 } )$ 
C)  $\ln ( 14 )$ 
D)  $\ln ( 7 - \sqrt { 48 } )$ 
A)  $\ln ( 7 + \sqrt { 48 } )$ 
B)  $\ln ( 7 + \sqrt { 50 } )$ 
C)  $\ln ( 14 )$ 
D)  $\ln ( 7 - \sqrt { 48 } )$

Question 14

Evaluate the integral.
-$$\int _ { - \pi / 4 } ^ { \pi / 4 } 3 \sinh ( \tan \theta ) \sec ^ { 2 } \theta d \theta$$

Accepted Answer

A)  3 
B)  0 
C)  $3 \left( e - \frac { 1 } { e } \right)$ 
D)  $3 \left( e ^ { \pi / 4 } - e ^ { - \pi / 4 } \right)$ 
A)  3 
B)  0 
C)  $3 \left( e - \frac { 1 } { e } \right)$ 
D)  $3 \left( e ^ { \pi / 4 } - e ^ { - \pi / 4 } \right)$

Question 15

Evaluate the integral.
-$$\int \operatorname { sech } ^ { 2 } ( 6 x - 10 ) d x$$

Accepted Answer

A)  $\frac { 1 } { 6 } \operatorname { coth } ( 6 x - 10 ) + C$ 
B)  $2 \operatorname { sech } ^ { 3 } ( 6 x - 10 ) + C$ 
C)  $\tanh ( 6 x - 10 ) + C$ 
D)  $\frac { 1 } { 6 } \tanh ( 6 x - 10 ) + C$ 
A)  $\frac { 1 } { 6 } \operatorname { coth } ( 6 x - 10 ) + C$ 
B)  $2 \operatorname { sech } ^ { 3 } ( 6 x - 10 ) + C$ 
C)  $\tanh ( 6 x - 10 ) + C$ 
D)  $\frac { 1 } { 6 } \tanh ( 6 x - 10 ) + C$

Question 16

Find the derivative of y.
-$$y = \operatorname { csch } \frac { 4 x } { 11 }$$

Accepted Answer

A)  $\frac { 4 } { 11 } \operatorname { csch } \frac { 4 x } { 11 } \operatorname { coth } \frac { 4 x } { 11 }$ 
B)  $- \frac { 4 } { 11 } \operatorname { csch } \frac { 4 x } { 11 } \operatorname { coth } \frac { 4 x } { 11 }$ 
C)  $- \operatorname { csch } \frac { 4 x } { 11 } \operatorname { coth } \frac { 4 x } { 11 }$ 
D)  $\operatorname { csch } \frac { 4 x } { 11 } \operatorname { coth } \frac { 4 x } { 11 }$ 
A)  $\frac { 4 } { 11 } \operatorname { csch } \frac { 4 x } { 11 } \operatorname { coth } \frac { 4 x } { 11 }$ 
B)  $- \frac { 4 } { 11 } \operatorname { csch } \frac { 4 x } { 11 } \operatorname { coth } \frac { 4 x } { 11 }$ 
C)  $- \operatorname { csch } \frac { 4 x } { 11 } \operatorname { coth } \frac { 4 x } { 11 }$ 
D)  $\operatorname { csch } \frac { 4 x } { 11 } \operatorname { coth } \frac { 4 x } { 11 }$

Question 17

Express the value of the inverse hyperbolic function in terms of natural logarithms.
-$\operatorname { sech } ^ { - 1 } \left( \frac { 12 } { 13 } \right)$

Accepted Answer

A)  $\ln \frac { 3 } { 2 }$ 
B)  Undefined 
C)  $\ln \frac { 18 } { 13 }$ 
D)  $\ln \frac { 2 } { 3 }$ 
A)  $\ln \frac { 3 } { 2 }$ 
B)  Undefined 
C)  $\ln \frac { 18 } { 13 }$ 
D)  $\ln \frac { 2 } { 3 }$

Question 18

Evaluate the integral.
-$\int _ { 0 } ^ { \pi / 2 } 5 ^ { \cos t } \sin t d t$

Accepted Answer

A)  $\frac { 5 ^ { \pi / 2 } - 1 } { \ln 5 }$ 
B)  4 
C)  $\frac { 4 } { \ln 5 }$ 
D)  $\frac { - 4 } { \ln 5 }$ 
A)  $\frac { 5 ^ { \pi / 2 } - 1 } { \ln 5 }$ 
B)  4 
C)  $\frac { 4 } { \ln 5 }$ 
D)  $\frac { - 4 } { \ln 5 }$

Question 19

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.
-$\cosh x = \frac { 17 } { 8 } , x < 0 , \operatorname { sech } x =$

Accepted Answer

A)  $\frac { 15 } { 17 }$ 
B)  $- \frac { 8 } { 15 }$ 
C)  $- \frac { 289 } { 64 }$ 
D)  $\frac { 8 } { 17 }$ 
A)  $\frac { 15 } { 17 }$ 
B)  $- \frac { 8 } { 15 }$ 
C)  $- \frac { 289 } { 64 }$ 
D)  $\frac { 8 } { 17 }$

Question 20

Solve the differential equation.
-$$\frac { d y } { d x } = 6 x \sqrt { 9 - y ^ { 2 } }$$

Accepted Answer

A)  $y = 3 \sin \left( 3 x ^ { 2 } + C \right)$ 
B)  $y = 3 \sin ^ { - 1 } \left( 3 x ^ { 2 } + C \right)$ 
C)  $y = \sin ^ { - 1 } \left( 3 x ^ { 2 } + C \right)$ 
D)  $y = \sin \left( 3 x ^ { 2 } + C \right)$ 
A)  $y = 3 \sin \left( 3 x ^ { 2 } + C \right)$ 
B)  $y = 3 \sin ^ { - 1 } \left( 3 x ^ { 2 } + C \right)$ 
C)  $y = \sin ^ { - 1 } \left( 3 x ^ { 2 } + C \right)$ 
D)  $y = \sin \left( 3 x ^ { 2 } + C \right)$

Rewrite the expression in terms of exponentials and simplify the results. - $\cosh 3 x + \sinh 3 x$

Evaluate the integral. - $\int \frac { 21 \mathrm { e } \sqrt { 7 x } } { 2 \sqrt { x } } \mathrm { dx }$

Rewrite the expression in terms of exponentials and simplify the results. - $( \sinh x + \cosh x ) ^ { 7 }$

Evaluate the integral. - $\int \frac { d x } { x ( 4 + 6 \ln x ) }$

Evaluate the integral in terms of an inverse hyperbolic function. - $\int _ { 1 / 7 } ^ { 2 / 11 } \frac { d x } { x \sqrt { 25 - 64 x ^ { 2 } } }$

Evaluate the integral. - $\int \frac { \operatorname { csch } ( \ln x ) \operatorname { coth } ( \ln x ) } { 4 x } d x$

Verify the integration formula. - $\int 2 x ^ { 3 } \operatorname { sech } ^ { - 1 } x ^ { 2 } d x = \frac { x ^ { 4 } } { 2 } \operatorname { sech } ^ { - 1 } x + \frac { 1 } { 2 } \sqrt { 1 - x ^ { 4 } } + C$

Rewrite the expression in terms of exponentials and simplify the results. - $\cosh 9 x - \sinh 9 x$

Rewrite the expression in terms of exponentials and simplify the results. - $\ln ( \cosh 5 x - \sinh 5 x ) + \ln ( \cosh 8 x + \sinh 8 x )$

Solve the initial value problem. - $\frac { d y } { d x } = - 5 e ^ { - x } \sec e ^ { - x } \tan e ^ { - x } , y ( 0 ) = 5 \sec 1 + 6$

Verify the integration formula. - $\int \operatorname { coth } ^ { - 1 } \sqrt { x } d x = ( x - 1 ) \operatorname { coth } ^ { - 1 } \sqrt { x } + \sqrt { x } + C$

Express the value of the inverse hyperbolic function in terms of natural logarithms. - $\cosh ^ { - 1 } 7$

Evaluate the integral. - $\int _ { - \pi / 4 } ^ { \pi / 4 } 3 \sinh ( \tan \theta ) \sec ^ { 2 } \theta d \theta$

Evaluate the integral. - $\int \operatorname { sech } ^ { 2 } ( 6 x - 10 ) d x$

Find the derivative of y. - $y = \operatorname { csch } \frac { 4 x } { 11 }$

Express the value of the inverse hyperbolic function in terms of natural logarithms. - $\operatorname { sech } ^ { - 1 } \left( \frac { 12 } { 13 } \right)$

Evaluate the integral. - $\int _ { 0 } ^ { \pi / 2 } 5 ^ { \cos t } \sin t d t$

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. - $\cosh x = \frac { 17 } { 8 } , x < 0 , \operatorname { sech } x =$

Solve the differential equation. - $\frac { d y } { d x } = 6 x \sqrt { 9 - y ^ { 2 } }$

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Exam 8: Integrals and Transcendental Functions

Rewrite the expression in terms of exponentials and simplify the results. - cosh⁡3x+sinh⁡3x\cosh 3 x + \sinh 3 xcosh3x+sinh3x

Evaluate the integral. - ∫21e7x2xdx\int \frac { 21 \mathrm { e } \sqrt { 7 x } } { 2 \sqrt { x } } \mathrm { dx }∫2x​21e7x​​dx

Rewrite the expression in terms of exponentials and simplify the results. - (sinh⁡x+cosh⁡x)7( \sinh x + \cosh x ) ^ { 7 }(sinhx+coshx)7

Evaluate the integral. - ∫dxx(4+6ln⁡x)\int \frac { d x } { x ( 4 + 6 \ln x ) }∫x(4+6lnx)dx​

Evaluate the integral in terms of an inverse hyperbolic function. - ∫1/72/11dxx25−64x2\int _ { 1 / 7 } ^ { 2 / 11 } \frac { d x } { x \sqrt { 25 - 64 x ^ { 2 } } }∫1/72/11​x25−64x2​dx​

Evaluate the integral. - ∫csch⁡(ln⁡x)coth⁡(ln⁡x)4xdx\int \frac { \operatorname { csch } ( \ln x ) \operatorname { coth } ( \ln x ) } { 4 x } d x∫4xcsch(lnx)coth(lnx)​dx

Rewrite the expression in terms of exponentials and simplify the results. - cosh⁡9x−sinh⁡9x\cosh 9 x - \sinh 9 xcosh9x−sinh9x

Rewrite the expression in terms of exponentials and simplify the results. - ln⁡(cosh⁡5x−sinh⁡5x)+ln⁡(cosh⁡8x+sinh⁡8x)\ln ( \cosh 5 x - \sinh 5 x ) + \ln ( \cosh 8 x + \sinh 8 x )ln(cosh5x−sinh5x)+ln(cosh8x+sinh8x)

Solve the initial value problem. - dydx=−5e−xsec⁡e−xtan⁡e−x,y(0)=5sec⁡1+6\frac { d y } { d x } = - 5 e ^ { - x } \sec e ^ { - x } \tan e ^ { - x } , y ( 0 ) = 5 \sec 1 + 6dxdy​=−5e−xsece−xtane−x,y(0)=5sec1+6

Verify the integration formula. - ∫coth⁡−1xdx=(x−1)coth⁡−1x+x+C\int \operatorname { coth } ^ { - 1 } \sqrt { x } d x = ( x - 1 ) \operatorname { coth } ^ { - 1 } \sqrt { x } + \sqrt { x } + C∫coth−1x​dx=(x−1)coth−1x​+x​+C

Express the value of the inverse hyperbolic function in terms of natural logarithms. - cosh⁡−17\cosh ^ { - 1 } 7cosh−17

Evaluate the integral. - ∫−π/4π/43sinh⁡(tan⁡θ)sec⁡2θdθ\int _ { - \pi / 4 } ^ { \pi / 4 } 3 \sinh ( \tan \theta ) \sec ^ { 2 } \theta d \theta∫−π/4π/4​3sinh(tanθ)sec2θdθ

Evaluate the integral. - ∫sech⁡2(6x−10)dx\int \operatorname { sech } ^ { 2 } ( 6 x - 10 ) d x∫sech2(6x−10)dx

Find the derivative of y. - y=csch⁡4x11y = \operatorname { csch } \frac { 4 x } { 11 }y=csch114x​

Express the value of the inverse hyperbolic function in terms of natural logarithms. - sech⁡−1(1213)\operatorname { sech } ^ { - 1 } \left( \frac { 12 } { 13 } \right)sech−1(1312​)

Evaluate the integral. - ∫0π/25cos⁡tsin⁡tdt\int _ { 0 } ^ { \pi / 2 } 5 ^ { \cos t } \sin t d t∫0π/2​5costsintdt

Solve the differential equation. - dydx=6x9−y2\frac { d y } { d x } = 6 x \sqrt { 9 - y ^ { 2 } }dxdy​=6x9−y2​