Question 1

Evaluate the integral.
-$$\int _ { 0 } ^ { \pi / 12 } 12 \tan 3 x d x$$

Accepted Answer

A)  -2 ln 2 
B)  2 ln 3 
C)  4 ln 2 
D)  2 ln 2 
A)  -2 ln 2 
B)  2 ln 3 
C)  4 ln 2 
D)  2 ln 2

Question 2

Find the derivative of y.
-$$y = \ln ( \sinh 6 x )$$

Accepted Answer

A)  $\frac { 1 } { \sinh 6 x }$ 
B)  $6 \operatorname { csch } 6 x$ 
C)  $\operatorname { coth } 6 x$ 
D)  $6 \operatorname { coth } 6 x$ 
A)  $\frac { 1 } { \sinh 6 x }$ 
B)  $6 \operatorname { csch } 6 x$ 
C)  $\operatorname { coth } 6 x$ 
D)  $6 \operatorname { coth } 6 x$

Question 3

Solve the differential equation.
-$$\frac { d y } { d x } = \frac { 8 y ^ { 2 } } { x }$$

Accepted Answer

A)  $y = \frac { 1 } { 8 \ln x + C }$ 
B)  $y = 24 \ln x + C$ 
C)  $y = \frac { - 1 } { 8 \ln x + C }$ 
D)  $y = - 8 \ln x + C$ 
A)  $y = \frac { 1 } { 8 \ln x + C }$ 
B)  $y = 24 \ln x + C$ 
C)  $y = \frac { - 1 } { 8 \ln x + C }$ 
D)  $y = - 8 \ln x + C$

Question 4

Evaluate the integral.
-$$\int \frac { \cos x d x } { 1 + 4 \sin x }$$

Accepted Answer

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

Question 5

Determine if the given function y = f(x) is a solution of the accompanying differential equation.
-$$\begin{array} { l } 
y ^ { \prime } = - y ^ { 2 } \
y = \frac { 1 } { x + 8 }
\end{array}$$

Accepted Answer

A) True 
 B)False

Question 6

Solve the problem.
-The amount of alcohol in the bloodstream, $\mathrm { A }$, declines at a rate proportional to the amount, that is, $\frac { \mathrm { dA } } { \mathrm { dt } } = - \mathrm { kA }$. If $\mathrm { k } = 0.5$ for a particular person, how long will it take for his alcohol concentration to decrease from $0.10 \%$ to $0.05 \%$ ? Give your answer to the nearest tenth of an hour.

Accepted Answer

A)  2.8 hr 
B)  0.3 hr 
C)  2.1 hr 
D)  1.4 hr 
A)  2.8 hr 
B)  0.3 hr 
C)  2.1 hr 
D)  1.4 hr

Question 7

Evaluate the integral in terms of natural logarithms.
-$$\int _ { 0 } ^ { 7 / 8 } \frac { d x } { 1 - x ^ { 2 } }$$

Accepted Answer

A)  $\ln 29$ 
B)  $\frac { 1 } { 2 } \ln 15$ 
C)  $\frac { 1 } { 2 } \ln \left( \frac { 1 } { 15 } \right)$ 
D)  $\ln - 6$ 
A)  $\ln 29$ 
B)  $\frac { 1 } { 2 } \ln 15$ 
C)  $\frac { 1 } { 2 } \ln \left( \frac { 1 } { 15 } \right)$ 
D)  $\ln - 6$

Question 8

Solve the differential equation.
-$$x ^ { 2 } \frac { d y } { d x } = 3 y$$

Accepted Answer

A)  $\mathrm { y } = \mathrm { Ce } ^ { 3 / \mathrm { x } }$ 
B)  $y = C e ^ { - 3 / x }$ 
C)  $y = - \frac { 3 } { x } + C$ 
D)  $\mathrm { y } = \mathrm { Ce } ^ { - 3 \mathrm { x } }$ 
A)  $\mathrm { y } = \mathrm { Ce } ^ { 3 / \mathrm { x } }$ 
B)  $y = C e ^ { - 3 / x }$ 
C)  $y = - \frac { 3 } { x } + C$ 
D)  $\mathrm { y } = \mathrm { Ce } ^ { - 3 \mathrm { x } }$

Question 9

Rewrite the expression in terms of exponentials and simplify the results.
-$\sinh ( 4 \ln x )$

Accepted Answer

A)  $\frac { 1 } { 2 } \left( x ^ { 4 } + \frac { 1 } { x ^ { 4 } } \right)$ 
B)  $\frac { 1 } { 2 } \left( x ^ { 4 } - \frac { 1 } { x ^ { 4 } } \right)$ 
C)  $2 \left( x - \frac { 1 } { x } \right)$ 
D)  $2 x$ 
A)  $\frac { 1 } { 2 } \left( x ^ { 4 } + \frac { 1 } { x ^ { 4 } } \right)$ 
B)  $\frac { 1 } { 2 } \left( x ^ { 4 } - \frac { 1 } { x ^ { 4 } } \right)$ 
C)  $2 \left( x - \frac { 1 } { x } \right)$ 
D)  $2 x$

Question 10

Evaluate the integral.
-$$\int 9 \cosh \left( \frac { x } { 2 } - \ln 7 \right) d x$$

Accepted Answer

A)  $18 \sinh \left( \frac { x } { 2 } - \ln 7 \right) + C$ 
B)  $\frac { 9 } { 2 } \sinh \left( \frac { x } { 2 } \right) + C$ 
C)  $9 \sinh \left( \frac { x } { 2 } - \ln 7 \right) + C$ 
D)  $\frac { 18 } { 7 } \sinh \left( \frac { x } { 2 } - \ln 7 \right) + C$ 
A)  $18 \sinh \left( \frac { x } { 2 } - \ln 7 \right) + C$ 
B)  $\frac { 9 } { 2 } \sinh \left( \frac { x } { 2 } \right) + C$ 
C)  $9 \sinh \left( \frac { x } { 2 } - \ln 7 \right) + C$ 
D)  $\frac { 18 } { 7 } \sinh \left( \frac { x } { 2 } - \ln 7 \right) + C$

Question 11

Determine if the given function y = f(x) is a solution of the accompanying differential equation.
-$$\begin{array} { l } 
y ^ { \prime } + y = e ^ { 2 x } \
y = 8 e ^ { - x } + e ^ { 2 x }
\end{array}$$

Accepted Answer

A) True 
 B)False

Question 12

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.
-$\sinh x = \frac { 8 } { 15 } , \operatorname { sech } x =$

Accepted Answer

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

Question 13

Solve the initial value problem.
-$$\frac { d y } { d x } = e ^ { 4 x } \cos e ^ { 4 x } , y ( 0 ) = 0$$

Accepted Answer

A)  $y = \frac { 1 } { 4 } \sin x$ 
B)  $y = \frac { 1 } { 4 } \sin \mathrm { e } ^ { 4 x } - \frac { 1 } { 4 } \sin 1$ 
C)  $y = - \frac { 1 } { 4 } \sin e ^ { 4 x } + \frac { 1 } { 4 } \sin 1$ 
D)  $y = \frac { 1 } { 4 } \sin \mathrm { e } ^ { 4 \mathrm { x } } - \frac { 1 } { 4 }$ 
A)  $y = \frac { 1 } { 4 } \sin x$ 
B)  $y = \frac { 1 } { 4 } \sin \mathrm { e } ^ { 4 x } - \frac { 1 } { 4 } \sin 1$ 
C)  $y = - \frac { 1 } { 4 } \sin e ^ { 4 x } + \frac { 1 } { 4 } \sin 1$ 
D)  $y = \frac { 1 } { 4 } \sin \mathrm { e } ^ { 4 \mathrm { x } } - \frac { 1 } { 4 }$

Question 14

Determine if the given function y = f(x) is a solution of the accompanying differential equation.
-Differential equation: $y ^ { \prime } = e ^ { - 5 x } - 5 y$ Initial condition: $y ( 0 ) = 0$
Solution candidate: $y = 5 x e ^ { - 5 x }$

Accepted Answer

A) True 
 B)False

Question 15

Evaluate the integral.
-$$\int \cosh \frac { x } { 7 } d x$$

Accepted Answer

A)  $\sinh \frac { x } { 7 } + C$ 
B)  $\sin ^ { - 1 } \frac { x } { 7 } + C$ 
C)  $7 \sinh \frac { x } { 7 } + C$ 
D)  $- 7 \sinh \frac { x } { 7 } + C$ 
A)  $\sinh \frac { x } { 7 } + C$ 
B)  $\sin ^ { - 1 } \frac { x } { 7 } + C$ 
C)  $7 \sinh \frac { x } { 7 } + C$ 
D)  $- 7 \sinh \frac { x } { 7 } + C$

Question 16

Evaluate the integral.
-$$\int _ { 2 } ^ { 3 } \frac { x ^ { 2 } + 1 } { x ^ { 3 } + 3 x } d x$$

Accepted Answer

A)  $\frac { 1 } { 3 } \ln \left| \frac { 18 } { 7 } \right|$ 
B)  $\frac { 1 } { 3 } \ln \left| \frac { 2 } { 3 } \right|$ 
C)  $\frac { 2 } { 3 } \ln \left| \frac { 3 } { 2 } \right|$ 
D)  $\frac { 1 } { 3 } \ln \left| \frac { 14 } { 33 } \right|$ 
A)  $\frac { 1 } { 3 } \ln \left| \frac { 18 } { 7 } \right|$ 
B)  $\frac { 1 } { 3 } \ln \left| \frac { 2 } { 3 } \right|$ 
C)  $\frac { 2 } { 3 } \ln \left| \frac { 3 } { 2 } \right|$ 
D)  $\frac { 1 } { 3 } \ln \left| \frac { 14 } { 33 } \right|$

Evaluate the integral. - $\int _ { 0 } ^ { \pi / 12 } 12 \tan 3 x d x$

Find the derivative of y. - $y = \ln ( \sinh 6 x )$

Solve the differential equation. - $\frac { d y } { d x } = \frac { 8 y ^ { 2 } } { x }$

Evaluate the integral. - $\int \frac { \cos x d x } { 1 + 4 \sin x }$

Determine if the given function y = f(x) is a solution of the accompanying differential equation. - =- y=

Evaluate the integral in terms of natural logarithms. - $\int _ { 0 } ^ { 7 / 8 } \frac { d x } { 1 - x ^ { 2 } }$

Solve the differential equation. - $x ^ { 2 } \frac { d y } { d x } = 3 y$

Rewrite the expression in terms of exponentials and simplify the results. - $\sinh ( 4 \ln x )$

Evaluate the integral. - $\int 9 \cosh \left( \frac { x } { 2 } - \ln 7 \right) d x$

Determine if the given function y = f(x) is a solution of the accompanying differential equation. - +y= y=8+

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. - $\sinh x = \frac { 8 } { 15 } , \operatorname { sech } x =$

Solve the initial value problem. - $\frac { d y } { d x } = e ^ { 4 x } \cos e ^ { 4 x } , y ( 0 ) = 0$

Determine if the given function y = f(x) is a solution of the accompanying differential equation. -Differential equation: $y ^ { \prime } = e ^ { - 5 x } - 5 y$ Initial condition: $y ( 0 ) = 0$ Solution candidate: $y = 5 x e ^ { - 5 x }$

Evaluate the integral. - $\int \cosh \frac { x } { 7 } d x$

Evaluate the integral. - $\int _ { 2 } ^ { 3 } \frac { x ^ { 2 } + 1 } { x ^ { 3 } + 3 x } d x$

Functions

Limits and Continuity

Derivatives

Applications of Derivatives

Integrals

Applications of Definite Integrals

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions and Motion in Space

Partial Derivatives

Multiple Integrals

Integrals and Vector Fields

Filters

Exam 8: Integrals and Transcendental Functions

Evaluate the integral. - ∫0π/1212tan⁡3xdx\int _ { 0 } ^ { \pi / 12 } 12 \tan 3 x d x∫0π/12​12tan3xdx

Find the derivative of y. - y=ln⁡(sinh⁡6x)y = \ln ( \sinh 6 x )y=ln(sinh6x)

Solve the differential equation. - dydx=8y2x\frac { d y } { d x } = \frac { 8 y ^ { 2 } } { x }dxdy​=x8y2​

Evaluate the integral. - ∫cos⁡xdx1+4sin⁡x\int \frac { \cos x d x } { 1 + 4 \sin x }∫1+4sinxcosxdx​

Determine if the given function y = f(x) is a solution of the accompanying differential equation. - =- y=

Evaluate the integral in terms of natural logarithms. - ∫07/8dx1−x2\int _ { 0 } ^ { 7 / 8 } \frac { d x } { 1 - x ^ { 2 } }∫07/8​1−x2dx​

Solve the differential equation. - x2dydx=3yx ^ { 2 } \frac { d y } { d x } = 3 yx2dxdy​=3y

Rewrite the expression in terms of exponentials and simplify the results. - sinh⁡(4ln⁡x)\sinh ( 4 \ln x )sinh(4lnx)

Evaluate the integral. - ∫9cosh⁡(x2−ln⁡7)dx\int 9 \cosh \left( \frac { x } { 2 } - \ln 7 \right) d x∫9cosh(2x​−ln7)dx

Determine if the given function y = f(x) is a solution of the accompanying differential equation. - +y= y=8+

Solve the initial value problem. - dydx=e4xcos⁡e4x,y(0)=0\frac { d y } { d x } = e ^ { 4 x } \cos e ^ { 4 x } , y ( 0 ) = 0dxdy​=e4xcose4x,y(0)=0

Determine if the given function y = f(x) is a solution of the accompanying differential equation. -Differential equation: y′=e−5x−5yy ^ { \prime } = e ^ { - 5 x } - 5 yy′=e−5x−5y Initial condition: y(0)=0y ( 0 ) = 0y(0)=0 Solution candidate: y=5xe−5xy = 5 x e ^ { - 5 x }y=5xe−5x

Evaluate the integral. - ∫cosh⁡x7dx\int \cosh \frac { x } { 7 } d x∫cosh7x​dx

Evaluate the integral. - ∫23x2+1x3+3xdx\int _ { 2 } ^ { 3 } \frac { x ^ { 2 } + 1 } { x ^ { 3 } + 3 x } d x∫23​x3+3xx2+1​dx

Functions

Limits and Continuity

Derivatives

Applications of Derivatives

Integrals

Applications of Definite Integrals

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions and Motion in Space

Partial Derivatives

Multiple Integrals

Integrals and Vector Fields

Filters

Evaluate the integral. - $\int _ { 0 } ^ { \pi / 12 } 12 \tan 3 x d x$

Find the derivative of y. - $y = \ln ( \sinh 6 x )$

Solve the differential equation. - $\frac { d y } { d x } = \frac { 8 y ^ { 2 } } { x }$

Evaluate the integral. - $\int \frac { \cos x d x } { 1 + 4 \sin x }$

Evaluate the integral in terms of natural logarithms. - $\int _ { 0 } ^ { 7 / 8 } \frac { d x } { 1 - x ^ { 2 } }$

Solve the differential equation. - $x ^ { 2 } \frac { d y } { d x } = 3 y$

Rewrite the expression in terms of exponentials and simplify the results. - $\sinh ( 4 \ln x )$

Evaluate the integral. - $\int 9 \cosh \left( \frac { x } { 2 } - \ln 7 \right) d x$

Solve the initial value problem. - $\frac { d y } { d x } = e ^ { 4 x } \cos e ^ { 4 x } , y ( 0 ) = 0$

Determine if the given function y = f(x) is a solution of the accompanying differential equation. -Differential equation: $y ^ { \prime } = e ^ { - 5 x } - 5 y$ Initial condition: $y ( 0 ) = 0$ Solution candidate: $y = 5 x e ^ { - 5 x }$

Evaluate the integral. - $\int \cosh \frac { x } { 7 } d x$

Evaluate the integral. - $\int _ { 2 } ^ { 3 } \frac { x ^ { 2 } + 1 } { x ^ { 3 } + 3 x } d x$