Question 1

Determine if the given function y = f(x) is a solution of the accompanying differential equation.
-$$\begin{array} { l } 
y ^ { \prime } - y = 3 e ^ { x } \
y = e ^ { 3 x } + 3 x e ^ { x }
\end{array}$$

Accepted Answer

A) True 
 B)False

Question 2

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

Accepted Answer

A)  $y = \ln \left( x ^ { 3 } + C \right)$ 
B)  $y = \ln \left( 3 x ^ { 3 } + C \right)$ 
C)  $y = C \ln \left( x ^ { 3 } \right)$ 
D)  $y = x ^ { 3 } + C$ 
A)  $y = \ln \left( x ^ { 3 } + C \right)$ 
B)  $y = \ln \left( 3 x ^ { 3 } + C \right)$ 
C)  $y = C \ln \left( x ^ { 3 } \right)$ 
D)  $y = x ^ { 3 } + C$

Question 3

Evaluate the integral.
-$$\int \frac { 8 d x } { 9 + 3 x }$$

Accepted Answer

A)  $\frac { 4 } { 3 } \ln | 9 + 3 x | + C$ 
B)  $\ln | - 9 - 3 x | + C$ 
C)  $- 8 \ln | - 9 - 3 x | + C$ 
D)  $\frac { 8 } { 3 } \ln | - 9 - 3 x | + C$ 
A)  $\frac { 4 } { 3 } \ln | 9 + 3 x | + C$ 
B)  $\ln | - 9 - 3 x | + C$ 
C)  $- 8 \ln | - 9 - 3 x | + C$ 
D)  $\frac { 8 } { 3 } \ln | - 9 - 3 x | + C$

Question 4

Find the derivative of y with respect to the appropriate variable.
-$$y = \cosh ^ { - 1 } 2 \sqrt { x + 3 }$$

Accepted Answer

A)  $\frac { 1 } { \sqrt { ( 2 x + 5 ) } }$ 
B)  $\frac { 1 } { \sqrt { ( 4 x + 13 ) ( x + 3 ) } }$ 
C)  $\frac { 1 } { \sqrt { ( 4 x + 11 ) ( x + 3 ) } }$ 
D)  $\frac { 1 } { \sqrt { ( 2 x + 5 ) ( x + 3 ) } }$ 
A)  $\frac { 1 } { \sqrt { ( 2 x + 5 ) } }$ 
B)  $\frac { 1 } { \sqrt { ( 4 x + 13 ) ( x + 3 ) } }$ 
C)  $\frac { 1 } { \sqrt { ( 4 x + 11 ) ( x + 3 ) } }$ 
D)  $\frac { 1 } { \sqrt { ( 2 x + 5 ) ( x + 3 ) } }$

Question 5

Express the value of the inverse hyperbolic function in terms of natural logarithms.
-$\tanh ^ { - 1 } \left( \frac { 9 } { 10 } \right)$

Accepted Answer

A)  $\frac { 1 } { 2 } \ln 209$ 
B)  $\frac { 1 } { 2 } \ln 1$ 
C)  $\frac { 1 } { 2 } \ln - 19$ 
D)  $\frac { 1 } { 2 } \ln 19$ 
A)  $\frac { 1 } { 2 } \ln 209$ 
B)  $\frac { 1 } { 2 } \ln 1$ 
C)  $\frac { 1 } { 2 } \ln - 19$ 
D)  $\frac { 1 } { 2 } \ln 19$

Question 6

Solve the problem.
-Find the volume of the solid that is generated by revolving the area bounded by the $x$-axis, the curve $y = \sqrt { \frac { 7 x } { x ^ { 2 } + 1 } } , x = 1$, and $x = 8$ about the $x$-axis.

Accepted Answer

A)  $7 \pi \ln \left( \frac { 65 } { 2 } \right)$ 
B)  $\frac { 7 } { 4 } \pi \ln \left( \frac { 65 } { 2 } \right)$ 
C)  $\frac { 7 } { 2 } \pi \ln \left( \frac { 65 } { 2 } \right)$ 
D)  $\frac { 7 } { 2 } \ln \left( \frac { 65 } { 2 } \right)$ 
A)  $7 \pi \ln \left( \frac { 65 } { 2 } \right)$ 
B)  $\frac { 7 } { 4 } \pi \ln \left( \frac { 65 } { 2 } \right)$ 
C)  $\frac { 7 } { 2 } \pi \ln \left( \frac { 65 } { 2 } \right)$ 
D)  $\frac { 7 } { 2 } \ln \left( \frac { 65 } { 2 } \right)$

Question 7

Evaluate the integral.
-$$\int _ { 1 } ^ { 2 } 9 x ^ { 2 } x ^ { 3 } d x$$

Accepted Answer

A)  $\frac { 19,674 } { \ln 3 }$ 
B)  $\frac { 3 } { \ln x } + C$ 
C)  $\frac { 72 } { \ln 3 }$ 
D)  19,674 
A)  $\frac { 19,674 } { \ln 3 }$ 
B)  $\frac { 3 } { \ln x } + C$ 
C)  $\frac { 72 } { \ln 3 }$ 
D)  19,674

Question 8

Provide an appropriate response.
-Show that $\lim _ { x \rightarrow } \frac { \ln ( x + 1 ) } { \ln x } = \lim _ { x \rightarrow } \frac { \ln ( x + 9982 ) } { \ln x }$. Explain why this is the case.

Accepted Answer

@#IMG-DLM&
@#IMG-DLM&

@#LAT-DLM&, and @#LAT-DLM&

Question 9

Solve the initial value problem.
-$$\frac { d y } { d t } = e ^ { - t } \sec ^ { 2 } \left( \pi e ^ { - t } \right) , y ( - \ln 5 ) = \frac { 1 } { \pi }$$

Accepted Answer

A)  $y = \frac { \tan \left( \pi \mathrm { e } ^ { - \mathrm { t } } \right) + 6 } { \pi }$ 
B)  $y = \frac { - e ^ { - t } \cot \left( \pi e ^ { - t } \right) + 0 } { \pi }$ 
C)  $y = \cot \left( \pi e ^ { - t } \right) + 1$ 
D)  $y = \frac { - \tan \left( \pi e ^ { - t } \right) + 1 } { \pi }$ 
A)  $y = \frac { \tan \left( \pi \mathrm { e } ^ { - \mathrm { t } } \right) + 6 } { \pi }$ 
B)  $y = \frac { - e ^ { - t } \cot \left( \pi e ^ { - t } \right) + 0 } { \pi }$ 
C)  $y = \cot \left( \pi e ^ { - t } \right) + 1$ 
D)  $y = \frac { - \tan \left( \pi e ^ { - t } \right) + 1 } { \pi }$

Question 10

Express the value of the inverse hyperbolic function in terms of natural logarithms.
-$\operatorname { coth } ^ { - 1 } \left( \frac { 6 } { 5 } \right)$

Accepted Answer

A)  $\frac { 1 } { 2 } \ln 11$ 
B)  $\frac { 1 } { 2 } \ln 66$ 
C)  0 
D)  $\frac { 1 } { 2 } \ln - 11$ 
A)  $\frac { 1 } { 2 } \ln 11$ 
B)  $\frac { 1 } { 2 } \ln 66$ 
C)  0 
D)  $\frac { 1 } { 2 } \ln - 11$

Question 11

Solve the initial value problem.
-$$\frac { \mathrm { d } ^ { 2 } y } { d x ^ { 2 } } = - 5 e ^ { - x } , y ( 0 ) = - 4 , y ^ { \prime } ( 0 ) = 0$$

Accepted Answer

A)  $y = - 5 e ^ { - x } - 4$ 
B)  $y = - 5 e ^ { - x } + 5 x - 9$ 
C)  $y = 5 e ^ { - x } + C$ 
D)  $y = - 5 e ^ { - x } - 5 x + 1$ 
A)  $y = - 5 e ^ { - x } - 4$ 
B)  $y = - 5 e ^ { - x } + 5 x - 9$ 
C)  $y = 5 e ^ { - x } + C$ 
D)  $y = - 5 e ^ { - x } - 5 x + 1$

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.
-$\cosh x = \frac { 13 } { 12 } , x > 0 , \tanh x =$

Accepted Answer

A)  $\frac { 5 } { 13 }$ 
B)  $\frac { 5 } { 12 }$ 
C)  $- \frac { 25 } { 144 }$ 
D)  $\frac { 13 } { 5 }$ 
A)  $\frac { 5 } { 13 }$ 
B)  $\frac { 5 } { 12 }$ 
C)  $- \frac { 25 } { 144 }$ 
D)  $\frac { 13 } { 5 }$

Question 13

Find the derivative of y with respect to the appropriate variable.
-$$y = \left( \theta ^ { 2 } + 4 \theta \right) \tanh ^ { - 1 } ( \theta + 3 )$$

Accepted Answer

A)  $( 2 \theta + 4 ) \tanh ^ { - 1 } ( \theta + 3 ) - \frac { \theta } { \theta + 2 }$ 
B)  $( 2 \theta + 4 ) - \frac { 1 } { \theta + 8 }$ 
C)  $( 2 \theta + 4 ) \tanh ^ { - 1 } ( \theta + 3 ) - \frac { \theta ^ { 2 } + 4 \theta } { 1 + ( \theta + 3 ) ^ { 2 } }$ 
D)  $- \frac { \theta } { \theta + 2 }$ 
A)  $( 2 \theta + 4 ) \tanh ^ { - 1 } ( \theta + 3 ) - \frac { \theta } { \theta + 2 }$ 
B)  $( 2 \theta + 4 ) - \frac { 1 } { \theta + 8 }$ 
C)  $( 2 \theta + 4 ) \tanh ^ { - 1 } ( \theta + 3 ) - \frac { \theta ^ { 2 } + 4 \theta } { 1 + ( \theta + 3 ) ^ { 2 } }$ 
D)  $- \frac { \theta } { \theta + 2 }$

Question 14

Evaluate the integral.
-$\int \tanh \left( \frac { x } { 2 } \right) d x$

Accepted Answer

A)  $2 \operatorname { sech } ^ { 2 } \frac { x } { 2 } + C$ 
B)  $2 \ln \left( \sinh \frac { x } { 2 } \right) + C$ 
C)  $\ln \left( \operatorname { coth } \frac { x } { 2 } \right) + C$ 
D)  $2 \ln \left( \cosh \frac { x } { 2 } \right) + C$ 
A)  $2 \operatorname { sech } ^ { 2 } \frac { x } { 2 } + C$ 
B)  $2 \ln \left( \sinh \frac { x } { 2 } \right) + C$ 
C)  $\ln \left( \operatorname { coth } \frac { x } { 2 } \right) + C$ 
D)  $2 \ln \left( \cosh \frac { x } { 2 } \right) + C$

Question 15

Solve the initial value problem.
-$$\frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dt } ^ { 2 } } = \mathrm { e } ^ { 2 \mathrm { t } } + 4 \sin \mathrm { t } , \mathrm { y } ( 0 ) = 0 , \mathrm { y } ^ { \prime } ( 0 ) = 8$$

Accepted Answer

A)  $y = \frac { e ^ { 2 t } } { 4 } - 4 \sin t + 8 t - \frac { 1 } { 4 }$ 
B)  $y = \frac { e ^ { 2 t } } { 4 } - 4 \sin t + \frac { 23 } { 2 } t - \frac { 1 } { 4 }$ 
C)  $y = e ^ { 2 t } - 4 \sin t + 11 t - \frac { 1 } { 4 }$ 
D)  $y = \frac { e ^ { 2 t } } { 4 } - 4 \sin t$ 
A)  $y = \frac { e ^ { 2 t } } { 4 } - 4 \sin t + 8 t - \frac { 1 } { 4 }$ 
B)  $y = \frac { e ^ { 2 t } } { 4 } - 4 \sin t + \frac { 23 } { 2 } t - \frac { 1 } { 4 }$ 
C)  $y = e ^ { 2 t } - 4 \sin t + 11 t - \frac { 1 } { 4 }$ 
D)  $y = \frac { e ^ { 2 t } } { 4 } - 4 \sin t$

Question 16

Determine if the given function y = f(x) is a solution of the accompanying differential equation.
-Differential equation: $x y ^ { \prime } + y = \frac { 2 x } { 9 }$
Initial condition: $y ( 9 ) = 2$
Solution candidate: $y = \frac { 9 } { x } + \frac { x } { 9 }$

Accepted Answer

A) True 
 B)False

Question 17

Find the slowest growing and the fastest growing functions as x→∞ .
-$$\begin{array} { l } 
y = 7 x ^ { 10 } \
y = e ^ { x } \
y = e ^ { x - 2 } \
y = x e ^ { x }
\end{array}$$

Accepted Answer

A)  Slowest: $y = 7 x ^ { 10 }$
Fastest: $y = e ^ { x }$ and $y = e ^ { x - 2 }$ grow at the same rate. 
B)  Slowest: $y = x e ^ { x }$
Fastest: $\mathrm { y } = \mathrm { e } ^ { \mathrm { x } }$ 
C)  Slowest: $y = e ^ { x - 2 }$
Fastest: $y = x e ^ { x }$ 
D)  Slowest: $\mathrm { y } = 7 \mathrm { x } ^ { 10 }$
Fastest: $y = x e ^ { x }$ 
A)  Slowest: $y = 7 x ^ { 10 }$
Fastest: $y = e ^ { x }$ and $y = e ^ { x - 2 }$ grow at the same rate. 
B)  Slowest: $y = x e ^ { x }$
Fastest: $\mathrm { y } = \mathrm { e } ^ { \mathrm { x } }$ 
C)  Slowest: $y = e ^ { x - 2 }$
Fastest: $y = x e ^ { x }$ 
D)  Slowest: $\mathrm { y } = 7 \mathrm { x } ^ { 10 }$
Fastest: $y = x e ^ { x }$

Question 18

Verify the integration formula.
-$$\int 2 \operatorname { sech } x d x = \sin ^ { - 1 } \left( 1 - x ^ { 2 } \right) + C$$

Accepted Answer

A) True 
 B)False

Question 19

Solve the problem.
-The region between the curve $y = \frac { 1 } { x ^ { 2 } }$ and the $x$-axis from $x = \frac { 1 } { 4 }$ to $x = 4$ is revolved about the $y$-axis to generate a solid. Find the volume of the solid.

Accepted Answer

A)  $2 \pi \ln 4 - \pi$ 
B)  $\pi \ln 4 - \pi$ 
C)  $4 \pi \ln 4$ 
D)  $2 \pi \ln 4$ 
A)  $2 \pi \ln 4 - \pi$ 
B)  $\pi \ln 4 - \pi$ 
C)  $4 \pi \ln 4$ 
D)  $2 \pi \ln 4$

Question 20

Find the derivative of y with respect to the appropriate variable.
-$y = 7 \tanh ^ { - 1 } ( \cos x )$

Accepted Answer

A)  $\frac { - 7 } { \sin x }$ 
B)  $\ln \left( \frac { 1 } { \sqrt { 1 - x ^ { 2 } } } \right) \sin x$ 
C)  $\frac { - 7 } { \cos x }$ 
D)  $\frac { - 7 \sin x } { 1 + \cos ^ { 2 } x }$ 
A)  $\frac { - 7 } { \sin x }$ 
B)  $\ln \left( \frac { 1 } { \sqrt { 1 - x ^ { 2 } } } \right) \sin x$ 
C)  $\frac { - 7 } { \cos x }$ 
D)  $\frac { - 7 \sin x } { 1 + \cos ^ { 2 } x }$

Determine if the given function y = f(x) is a solution of the accompanying differential equation. - -y=3 y=+3x

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

Evaluate the integral. - $\int \frac { 8 d x } { 9 + 3 x }$

Find the derivative of y with respect to the appropriate variable. - $y = \cosh ^ { - 1 } 2 \sqrt { x + 3 }$

Express the value of the inverse hyperbolic function in terms of natural logarithms. - $\tanh ^ { - 1 } \left( \frac { 9 } { 10 } \right)$

Solve the problem. -Find the volume of the solid that is generated by revolving the area bounded by the $x$ -axis, the curve $y = \sqrt { \frac { 7 x } { x ^ { 2 } + 1 } } , x = 1$ , and $x = 8$ about the $x$ -axis.

Evaluate the integral. - $\int _ { 1 } ^ { 2 } 9 x ^ { 2 } x ^ { 3 } d x$

Provide an appropriate response. -Show that $\lim _ { x \rightarrow } \frac { \ln ( x + 1 ) } { \ln x } = \lim _ { x \rightarrow } \frac { \ln ( x + 9982 ) } { \ln x }$ . Explain why this is the case.

Solve the initial value problem. - $\frac { d y } { d t } = e ^ { - t } \sec ^ { 2 } \left( \pi e ^ { - t } \right) , y ( - \ln 5 ) = \frac { 1 } { \pi }$

Express the value of the inverse hyperbolic function in terms of natural logarithms. - $\operatorname { coth } ^ { - 1 } \left( \frac { 6 } { 5 } \right)$

Solve the initial value problem. - $\frac { \mathrm { d } ^ { 2 } y } { d x ^ { 2 } } = - 5 e ^ { - x } , y ( 0 ) = - 4 , y ^ { \prime } ( 0 ) = 0$

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. - $\cosh x = \frac { 13 } { 12 } , x > 0 , \tanh x =$

Find the derivative of y with respect to the appropriate variable. - $y = \left( \theta ^ { 2 } + 4 \theta \right) \tanh ^ { - 1 } ( \theta + 3 )$

Evaluate the integral. - $\int \tanh \left( \frac { x } { 2 } \right) d x$

Solve the initial value problem. - $\frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dt } ^ { 2 } } = \mathrm { e } ^ { 2 \mathrm { t } } + 4 \sin \mathrm { t } , \mathrm { y } ( 0 ) = 0 , \mathrm { y } ^ { \prime } ( 0 ) = 8$

Determine if the given function y = f(x) is a solution of the accompanying differential equation. -Differential equation: $x y ^ { \prime } + y = \frac { 2 x } { 9 }$ Initial condition: $y ( 9 ) = 2$ Solution candidate: $y = \frac { 9 } { x } + \frac { x } { 9 }$

Find the slowest growing and the fastest growing functions as x→∞ . - y=7 y= y= y=x

Verify the integration formula. - $\int 2 \operatorname { sech } x d x = \sin ^ { - 1 } \left( 1 - x ^ { 2 } \right) + C$

Solve the problem. -The region between the curve $y = \frac { 1 } { x ^ { 2 } }$ and the $x$ -axis from $x = \frac { 1 } { 4 }$ to $x = 4$ is revolved about the $y$ -axis to generate a solid. Find the volume of the solid.

Find the derivative of y with respect to the appropriate variable. - $y = 7 \tanh ^ { - 1 } ( \cos x )$

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

Determine if the given function y = f(x) is a solution of the accompanying differential equation. - -y=3 y=+3x

Solve the differential equation. - dydx=3x2e−y\frac { d y } { d x } = 3 x ^ { 2 } e ^ { - y }dxdy​=3x2e−y

Evaluate the integral. - ∫8dx9+3x\int \frac { 8 d x } { 9 + 3 x }∫9+3x8dx​

Find the derivative of y with respect to the appropriate variable. - y=cosh⁡−12x+3y = \cosh ^ { - 1 } 2 \sqrt { x + 3 }y=cosh−12x+3​

Express the value of the inverse hyperbolic function in terms of natural logarithms. - tanh⁡−1(910)\tanh ^ { - 1 } \left( \frac { 9 } { 10 } \right)tanh−1(109​)

Solve the problem. -Find the volume of the solid that is generated by revolving the area bounded by the xxx -axis, the curve y=7xx2+1,x=1y = \sqrt { \frac { 7 x } { x ^ { 2 } + 1 } } , x = 1y=x2+17x​​,x=1 , and x=8x = 8x=8 about the xxx -axis.

Evaluate the integral. - ∫129x2x3dx\int _ { 1 } ^ { 2 } 9 x ^ { 2 } x ^ { 3 } d x∫12​9x2x3dx

Solve the initial value problem. - dydt=e−tsec⁡2(πe−t),y(−ln⁡5)=1π\frac { d y } { d t } = e ^ { - t } \sec ^ { 2 } \left( \pi e ^ { - t } \right) , y ( - \ln 5 ) = \frac { 1 } { \pi }dtdy​=e−tsec2(πe−t),y(−ln5)=π1​

Express the value of the inverse hyperbolic function in terms of natural logarithms. - coth⁡−1(65)\operatorname { coth } ^ { - 1 } \left( \frac { 6 } { 5 } \right)coth−1(56​)

Solve the initial value problem. - d2ydx2=−5e−x,y(0)=−4,y′(0)=0\frac { \mathrm { d } ^ { 2 } y } { d x ^ { 2 } } = - 5 e ^ { - x } , y ( 0 ) = - 4 , y ^ { \prime } ( 0 ) = 0dx2d2y​=−5e−x,y(0)=−4,y′(0)=0

Find the derivative of y with respect to the appropriate variable. - y=(θ2+4θ)tanh⁡−1(θ+3)y = \left( \theta ^ { 2 } + 4 \theta \right) \tanh ^ { - 1 } ( \theta + 3 )y=(θ2+4θ)tanh−1(θ+3)

Evaluate the integral. - ∫tanh⁡(x2)dx\int \tanh \left( \frac { x } { 2 } \right) d x∫tanh(2x​)dx

Find the slowest growing and the fastest growing functions as x→∞ . - y=7 y= y= y=x

Verify the integration formula. - ∫2sech⁡xdx=sin⁡−1(1−x2)+C\int 2 \operatorname { sech } x d x = \sin ^ { - 1 } \left( 1 - x ^ { 2 } \right) + C∫2sechxdx=sin−1(1−x2)+C

Solve the problem. -The region between the curve y=1x2y = \frac { 1 } { x ^ { 2 } }y=x21​ and the xxx -axis from x=14x = \frac { 1 } { 4 }x=41​ to x=4x = 4x=4 is revolved about the yyy -axis to generate a solid. Find the volume of the solid.

Find the derivative of y with respect to the appropriate variable. - y=7tanh⁡−1(cos⁡x)y = 7 \tanh ^ { - 1 } ( \cos x )y=7tanh−1(cosx)

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

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

Evaluate the integral. - $\int \frac { 8 d x } { 9 + 3 x }$

Find the derivative of y with respect to the appropriate variable. - $y = \cosh ^ { - 1 } 2 \sqrt { x + 3 }$

Express the value of the inverse hyperbolic function in terms of natural logarithms. - $\tanh ^ { - 1 } \left( \frac { 9 } { 10 } \right)$

Solve the problem. -Find the volume of the solid that is generated by revolving the area bounded by the $x$ -axis, the curve $y = \sqrt { \frac { 7 x } { x ^ { 2 } + 1 } } , x = 1$ , and $x = 8$ about the $x$ -axis.

Evaluate the integral. - $\int _ { 1 } ^ { 2 } 9 x ^ { 2 } x ^ { 3 } d x$

Solve the initial value problem. - $\frac { d y } { d t } = e ^ { - t } \sec ^ { 2 } \left( \pi e ^ { - t } \right) , y ( - \ln 5 ) = \frac { 1 } { \pi }$

Express the value of the inverse hyperbolic function in terms of natural logarithms. - $\operatorname { coth } ^ { - 1 } \left( \frac { 6 } { 5 } \right)$

Solve the initial value problem. - $\frac { \mathrm { d } ^ { 2 } y } { d x ^ { 2 } } = - 5 e ^ { - x } , y ( 0 ) = - 4 , y ^ { \prime } ( 0 ) = 0$

Find the derivative of y with respect to the appropriate variable. - $y = \left( \theta ^ { 2 } + 4 \theta \right) \tanh ^ { - 1 } ( \theta + 3 )$

Evaluate the integral. - $\int \tanh \left( \frac { x } { 2 } \right) d x$

Verify the integration formula. - $\int 2 \operatorname { sech } x d x = \sin ^ { - 1 } \left( 1 - x ^ { 2 } \right) + C$

Solve the problem. -The region between the curve $y = \frac { 1 } { x ^ { 2 } }$ and the $x$ -axis from $x = \frac { 1 } { 4 }$ to $x = 4$ is revolved about the $y$ -axis to generate a solid. Find the volume of the solid.

Find the derivative of y with respect to the appropriate variable. - $y = 7 \tanh ^ { - 1 } ( \cos x )$