Question 1

Solve the initial value problem.
-$$\frac { d y } { d t } = e ^ { t } \sin \left( e ^ { t } - 7 \right) , y ( \ln 7 ) = 0$$

Accepted Answer

A)  $y = \sin \mathrm { e } ^ { \mathrm { t } } - \sin 2$ 
B)  $y = \cos \left( e ^ { t } - 7 \right) - 1$ 
C)  $y = - \cos \left( e ^ { t } - 7 \right) + 1$ 
D)  $y = e ^ { t } \cos \left( e ^ { t } - 7 \right) - 7$ 
A)  $y = \sin \mathrm { e } ^ { \mathrm { t } } - \sin 2$ 
B)  $y = \cos \left( e ^ { t } - 7 \right) - 1$ 
C)  $y = - \cos \left( e ^ { t } - 7 \right) + 1$ 
D)  $y = e ^ { t } \cos \left( e ^ { t } - 7 \right) - 7$

Question 2

Provide an appropriate response.
-A polynomial $f ( x )$ has a degree smaller than or equal to another polynomial $g ( x )$. Does $f = O ( g )$ and does $\mathrm { g } = \mathrm { O } ( \mathrm { f } )$ ?

Accepted Answer

The answer of Provide an appropriate response.
-A polynomial \(f (...

Question 3

Solve the problem.
-A certain radioactive isotope decays at a rate of $2 \%$ per 100 years. If $t$ represents time in years and y represents the amount of the isotope left then the equation for the situation is $y = y 0 e ^ { - 0.0002 t }$. In how many years will there be $89 \%$ of the isotope left?

Accepted Answer

A)  550 years 
B)  583 years 
C)  1100 years 
D)  244 years 
A)  550 years 
B)  583 years 
C)  1100 years 
D)  244 years

Question 4

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

Accepted Answer

A)  $\frac { \ln 7 ( \ln x ) ^ { 2 } } { 2 } + C$ 
B)  $7 ^ { x } \ln 7 + C$ 
C)  $\frac { \ln x } { \ln 7 } + C$ 
D)  $\frac { ( \ln x ) ^ { 2 } } { 2 \ln 7 } + C$ 
A)  $\frac { \ln 7 ( \ln x ) ^ { 2 } } { 2 } + C$ 
B)  $7 ^ { x } \ln 7 + C$ 
C)  $\frac { \ln x } { \ln 7 } + C$ 
D)  $\frac { ( \ln x ) ^ { 2 } } { 2 \ln 7 } + C$

Question 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.
-$\cosh x = \frac { 13 } { 5 } , x < 0 , \operatorname { coth } x =$

Accepted Answer

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

Question 6

Solve the differential equation.
-$$\frac { d y } { d x } = 8 x ^ { 7 } \sec y$$

Accepted Answer

A)  $y = \sin ^ { - 1 } \left( x ^ { 8 } + C \right)$ 
B)  $y = \sin \left( x ^ { 8 } + C \right)$ 
C)  $y = \cos ^ { - 1 } \left( x ^ { 8 } + C \right)$ 
D)  $y = x ^ { 8 } + C$ 
A)  $y = \sin ^ { - 1 } \left( x ^ { 8 } + C \right)$ 
B)  $y = \sin \left( x ^ { 8 } + C \right)$ 
C)  $y = \cos ^ { - 1 } \left( x ^ { 8 } + C \right)$ 
D)  $y = x ^ { 8 } + C$

Question 7

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

Accepted Answer

A)  $5 \operatorname { sech } x ^ { 2 } + C$ 
B)  $\frac { \operatorname { sech } x ^ { 2 } } { 2 x } + C$ 
C)  $10 \operatorname { csch } x ^ { 2 } + C$ 
D)  $- 5 \operatorname { sech } x ^ { 2 } + C$ 
A)  $5 \operatorname { sech } x ^ { 2 } + C$ 
B)  $\frac { \operatorname { sech } x ^ { 2 } } { 2 x } + C$ 
C)  $10 \operatorname { csch } x ^ { 2 } + C$ 
D)  $- 5 \operatorname { sech } x ^ { 2 } + C$

Question 8

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

Accepted Answer

A) True 
 B)False

Question 9

Solve the problem.
-Find the length of the segment of the curve $y = \frac { 1 } { 2 } \cosh 2 x$ from $x = 0$ to $x = \ln \sqrt { 5 }$.

Accepted Answer

A)  $\frac { 13 } { 10 }$ 
B)  $\frac { 6 } { 5 }$ 
C)  5 
D)  $\frac { 1 } { 4 } \left( \sqrt { 5 } - \frac { 1 } { \sqrt { 5 } } \right)$ 
A)  $\frac { 13 } { 10 }$ 
B)  $\frac { 6 } { 5 }$ 
C)  5 
D)  $\frac { 1 } { 4 } \left( \sqrt { 5 } - \frac { 1 } { \sqrt { 5 } } \right)$

Question 10

Evaluate the integral.
-$$\int \frac { e ^ { 2 \theta } } { 1 + e ^ { 2 \theta } } d \theta$$

Accepted Answer

A)  $2 \ln \left( 1 + \mathrm { e } ^ { 2 \theta } \right) + \mathrm { C }$ 
B)  $\ln \left( 1 + \mathrm { e } ^ { 2 \theta } \right) + C$ 
C)  $\frac { \ln \left( 1 + 2 e ^ { \theta } \right) } { 2 } + C$ 
D)  $\frac { \ln \left( 1 + \mathrm { e } ^ { 2 \theta } \right) } { 2 } + \mathrm { C }$ 
A)  $2 \ln \left( 1 + \mathrm { e } ^ { 2 \theta } \right) + \mathrm { C }$ 
B)  $\ln \left( 1 + \mathrm { e } ^ { 2 \theta } \right) + C$ 
C)  $\frac { \ln \left( 1 + 2 e ^ { \theta } \right) } { 2 } + C$ 
D)  $\frac { \ln \left( 1 + \mathrm { e } ^ { 2 \theta } \right) } { 2 } + \mathrm { C }$

Question 11

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

Accepted Answer

A) True 
 B)False

Question 12

Express the value of the inverse hyperbolic function in terms of natural logarithms.
-$\operatorname { csch } ^ { - 1 } \left( \frac { 9 } { 4 } \right)$

Accepted Answer

A)  $\ln \left( \frac { 4 - \sqrt { 97 } } { 9 } \right)$ 
B)  $\ln \left( \frac { 4 + \sqrt { 97 } } { 9 } \right)$ 
C)  $\ln \frac { 3 } { 2 }$ 
D)  $\ln \sqrt { 97 }$ 
A)  $\ln \left( \frac { 4 - \sqrt { 97 } } { 9 } \right)$ 
B)  $\ln \left( \frac { 4 + \sqrt { 97 } } { 9 } \right)$ 
C)  $\ln \frac { 3 } { 2 }$ 
D)  $\ln \sqrt { 97 }$

Question 13

Determine if the given function y = f(x) is a solution of the accompanying differential equation.
-$$\begin{array} { l } 
y ^ { \prime } + y = e ^ { 2 x } \
y = 7 e ^ { - x } + \frac { 1 } { 3 } e ^ { 2 x }
\end{array}$$

Accepted Answer

A) True 
 B)False

Question 14

Evaluate the integral in terms of an inverse hyperbolic function.
-$$\int _ { \sqrt { 5 } } ^ { 6 \sqrt { 5 } } \frac { d x } { \sqrt { 36 + x ^ { 2 } } }$$

Accepted Answer

A)  $\sinh ^ { - 1 } ( \sqrt { 5 } ) - \sinh ^ { - 1 } \left( \frac { \sqrt { 5 } } { 6 } \right)$ 
B)  $- \frac { 1 } { 6 } \operatorname { csch } ^ { - 1 } ( \sqrt { 5 } ) - \frac { 1 } { 6 } \operatorname { csch } ^ { - 1 } \left( \frac { \sqrt { 5 } } { 6 } \right)$ 
C)  $\cosh ^ { - 1 } ( \sqrt { 5 } ) - \cosh ^ { - 1 } \left( \frac { \sqrt { 5 } } { 6 } \right)$ 
D)  $- \frac { 1 } { 6 } \operatorname { sech } ^ { - 1 } ( \sqrt { 5 } ) + \frac { 1 } { 6 } \operatorname { sech } ^ { - 1 } \left( \frac { \sqrt { 5 } } { 6 } \right)$ 
A)  $\sinh ^ { - 1 } ( \sqrt { 5 } ) - \sinh ^ { - 1 } \left( \frac { \sqrt { 5 } } { 6 } \right)$ 
B)  $- \frac { 1 } { 6 } \operatorname { csch } ^ { - 1 } ( \sqrt { 5 } ) - \frac { 1 } { 6 } \operatorname { csch } ^ { - 1 } \left( \frac { \sqrt { 5 } } { 6 } \right)$ 
C)  $\cosh ^ { - 1 } ( \sqrt { 5 } ) - \cosh ^ { - 1 } \left( \frac { \sqrt { 5 } } { 6 } \right)$ 
D)  $- \frac { 1 } { 6 } \operatorname { sech } ^ { - 1 } ( \sqrt { 5 } ) + \frac { 1 } { 6 } \operatorname { sech } ^ { - 1 } \left( \frac { \sqrt { 5 } } { 6 } \right)$

Question 15

Evaluate the integral.
-$$\int \frac { \mathrm { e } ^ { 1 / x } } { 5 x ^ { 2 } } d x$$

Accepted Answer

A)  $\frac { e ^ { - 1 / x } } { 5 } + C$ 
B)  $- 5 \mathrm { e } ^ { 1 / x } + C$ 
C)  $\frac { \mathrm { e } ^ { 1 / \mathrm { x } } } { 5 } + \mathrm { C }$ 
D)  $- \frac { e ^ { 1 / x } } { 5 } + C$ 
A)  $\frac { e ^ { - 1 / x } } { 5 } + C$ 
B)  $- 5 \mathrm { e } ^ { 1 / x } + C$ 
C)  $\frac { \mathrm { e } ^ { 1 / \mathrm { x } } } { 5 } + \mathrm { C }$ 
D)  $- \frac { e ^ { 1 / x } } { 5 } + C$

Question 16

Express the value of the inverse hyperbolic function in terms of natural logarithms.
-$$\sinh ^ { - 1 } \left( \frac { - 3 } { 4 } \right)$$

Accepted Answer

A)  $\ln \left( \frac { - 3 } { 4 } + \sqrt { 10 } \right)$ 
B)  $\ln ( 4 )$ 
C)  $\ln 2$ 
D)  $\ln \frac { 1 } { 2 }$ 
A)  $\ln \left( \frac { - 3 } { 4 } + \sqrt { 10 } \right)$ 
B)  $\ln ( 4 )$ 
C)  $\ln 2$ 
D)  $\ln \frac { 1 } { 2 }$

Question 17

Evaluate the integral in terms of natural logarithms.
-$$\int _ { 4 } ^ { 7 } \frac { d x } { \sqrt { x ^ { 2 } - 9 } }$$

Accepted Answer

A)  $\ln \left( \frac { 7 + \sqrt { 40 } } { 23 } \right) - 1$ 
B)  $\frac { 1 } { 2 } \ln \left( \frac { \sqrt { 40 } } { 4 + \sqrt { 7 } } \right)$ 
C)  $\ln 4$ 
D)  $\ln \left( \frac { 7 + \sqrt { 40 } } { 4 + \sqrt { 7 } } \right)$ 
A)  $\ln \left( \frac { 7 + \sqrt { 40 } } { 23 } \right) - 1$ 
B)  $\frac { 1 } { 2 } \ln \left( \frac { \sqrt { 40 } } { 4 + \sqrt { 7 } } \right)$ 
C)  $\ln 4$ 
D)  $\ln \left( \frac { 7 + \sqrt { 40 } } { 4 + \sqrt { 7 } } \right)$

Question 18

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.
-$\sinh x = \frac { 3 } { 4 } , \operatorname { coth } x =$

Accepted Answer

A)  $\frac { 3 } { 5 }$ 
B)  $\frac { 5 } { 16 }$ 
C)  $\frac { 5 } { 3 }$ 
D)  $\frac { 5 } { 4 }$ 
A)  $\frac { 3 } { 5 }$ 
B)  $\frac { 5 } { 16 }$ 
C)  $\frac { 5 } { 3 }$ 
D)  $\frac { 5 } { 4 }$

Question 19

Evaluate the integral in terms of natural logarithms.
-$$\int _ { 1 } ^ { e ^ { 5 } } \frac { 10 d x } { x \sqrt { 1 + ( \ln x ) ^ { 2 } } }$$

Accepted Answer

A)  5 
B)  $10 \ln ( 1 + \sqrt { 2 } )$ 
C)  $\ln ( 5 + \sqrt { 26 } )$ 
D)  $10 \ln ( 5 + \sqrt { 26 } )$ 
A)  5 
B)  $10 \ln ( 1 + \sqrt { 2 } )$ 
C)  $\ln ( 5 + \sqrt { 26 } )$ 
D)  $10 \ln ( 5 + \sqrt { 26 } )$

Question 20

Find the derivative of y.
-$y = \operatorname { sech } ( 5 \theta ) ( 1 - \ln \operatorname { sech } ( 5 \theta ) )$

Accepted Answer

A)  $1 - 5 \operatorname { sech } ( 5 \theta ) \tanh ( 5 \theta ) \ln \operatorname { sech } ( 5 \theta )$ 
B)  $5 \operatorname { sech } ( 5 \theta ) \tanh ( 5 \theta ) \ln ( 5 \theta )$ 
C)  $\operatorname { sech } ( 5 \theta ) \tanh ( 5 \theta ) \ln \operatorname { sech } ( 5 \theta )$ 
D)  $5 \operatorname { sech } ( 5 \theta ) \tanh ( 5 \theta ) \ln \operatorname { sech } ( 5 \theta )$ 
A)  $1 - 5 \operatorname { sech } ( 5 \theta ) \tanh ( 5 \theta ) \ln \operatorname { sech } ( 5 \theta )$ 
B)  $5 \operatorname { sech } ( 5 \theta ) \tanh ( 5 \theta ) \ln ( 5 \theta )$ 
C)  $\operatorname { sech } ( 5 \theta ) \tanh ( 5 \theta ) \ln \operatorname { sech } ( 5 \theta )$ 
D)  $5 \operatorname { sech } ( 5 \theta ) \tanh ( 5 \theta ) \ln \operatorname { sech } ( 5 \theta )$

Solve the initial value problem. - $\frac { d y } { d t } = e ^ { t } \sin \left( e ^ { t } - 7 \right) , y ( \ln 7 ) = 0$

Provide an appropriate response. -A polynomial $f ( x )$ has a degree smaller than or equal to another polynomial $g ( x )$ . Does $f = O ( g )$ and does $\mathrm { g } = \mathrm { O } ( \mathrm { f } )$ ?

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

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. - $\cosh x = \frac { 13 } { 5 } , x < 0 , \operatorname { coth } x =$

Solve the differential equation. - $\frac { d y } { d x } = 8 x ^ { 7 } \sec y$

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

Determine if the given function y = f(x) is a solution of the accompanying differential equation. - -y=2 y=+2x

Solve the problem. -Find the length of the segment of the curve $y = \frac { 1 } { 2 } \cosh 2 x$ from $x = 0$ to $x = \ln \sqrt { 5 }$ .

Evaluate the integral. - $\int \frac { e ^ { 2 \theta } } { 1 + e ^ { 2 \theta } } d \theta$

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

Express the value of the inverse hyperbolic function in terms of natural logarithms. - $\operatorname { csch } ^ { - 1 } \left( \frac { 9 } { 4 } \right)$

Determine if the given function y = f(x) is a solution of the accompanying differential equation. - +y= y=7+

Evaluate the integral in terms of an inverse hyperbolic function. - $\int _ { \sqrt { 5 } } ^ { 6 \sqrt { 5 } } \frac { d x } { \sqrt { 36 + x ^ { 2 } } }$

Evaluate the integral. - $\int \frac { \mathrm { e } ^ { 1 / x } } { 5 x ^ { 2 } } d x$

Express the value of the inverse hyperbolic function in terms of natural logarithms. - $\sinh ^ { - 1 } \left( \frac { - 3 } { 4 } \right)$

Evaluate the integral in terms of natural logarithms. - $\int _ { 4 } ^ { 7 } \frac { d x } { \sqrt { x ^ { 2 } - 9 } }$

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. - $\sinh x = \frac { 3 } { 4 } , \operatorname { coth } x =$

Evaluate the integral in terms of natural logarithms. - $\int _ { 1 } ^ { e ^ { 5 } } \frac { 10 d x } { x \sqrt { 1 + ( \ln x ) ^ { 2 } } }$

Find the derivative of y. - $y = \operatorname { sech } ( 5 \theta ) ( 1 - \ln \operatorname { sech } ( 5 \theta ) )$

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

Solve the initial value problem. - dydt=etsin⁡(et−7),y(ln⁡7)=0\frac { d y } { d t } = e ^ { t } \sin \left( e ^ { t } - 7 \right) , y ( \ln 7 ) = 0dtdy​=etsin(et−7),y(ln7)=0

Provide an appropriate response. -A polynomial f(x)f ( x )f(x) has a degree smaller than or equal to another polynomial g(x)g ( x )g(x) . Does f=O(g)f = O ( g )f=O(g) and does g=O(f)\mathrm { g } = \mathrm { O } ( \mathrm { f } )g=O(f) ?

Evaluate the integral. - ∫log⁡7xxdx\int \frac { \log 7 x } { x } d x∫xlog7x​dx

Solve the differential equation. - dydx=8x7sec⁡y\frac { d y } { d x } = 8 x ^ { 7 } \sec ydxdy​=8x7secy

Evaluate the integral. - ∫10xsech⁡x2tanh⁡x2dx\int 10 x \operatorname { sech } x ^ { 2 } \tanh x ^ { 2 } d x∫10xsechx2tanhx2dx

Determine if the given function y = f(x) is a solution of the accompanying differential equation. - -y=2 y=+2x

Solve the problem. -Find the length of the segment of the curve y=12cosh⁡2xy = \frac { 1 } { 2 } \cosh 2 xy=21​cosh2x from x=0x = 0x=0 to x=ln⁡5x = \ln \sqrt { 5 }x=ln5​ .

Evaluate the integral. - ∫e2θ1+e2θdθ\int \frac { e ^ { 2 \theta } } { 1 + e ^ { 2 \theta } } d \theta∫1+e2θe2θ​dθ

Verify the integration formula. - ∫xcsch⁡−1xdx=x22csch⁡−1x+121+x2+C\int x \operatorname { csch } ^ { - 1 } x d x = \frac { x ^ { 2 } } { 2 } \operatorname { csch } ^ { - 1 } x + \frac { 1 } { 2 } \sqrt { 1 + x ^ { 2 } } + C∫xcsch−1xdx=2x2​csch−1x+21​1+x2​+C

Express the value of the inverse hyperbolic function in terms of natural logarithms. - csch⁡−1(94)\operatorname { csch } ^ { - 1 } \left( \frac { 9 } { 4 } \right)csch−1(49​)

Determine if the given function y = f(x) is a solution of the accompanying differential equation. - +y= y=7+

Evaluate the integral in terms of an inverse hyperbolic function. - ∫565dx36+x2\int _ { \sqrt { 5 } } ^ { 6 \sqrt { 5 } } \frac { d x } { \sqrt { 36 + x ^ { 2 } } }∫5​65​​36+x2​dx​

Evaluate the integral. - ∫e1/x5x2dx\int \frac { \mathrm { e } ^ { 1 / x } } { 5 x ^ { 2 } } d x∫5x2e1/x​dx

Express the value of the inverse hyperbolic function in terms of natural logarithms. - sinh⁡−1(−34)\sinh ^ { - 1 } \left( \frac { - 3 } { 4 } \right)sinh−1(4−3​)

Evaluate the integral in terms of natural logarithms. - ∫47dxx2−9\int _ { 4 } ^ { 7 } \frac { d x } { \sqrt { x ^ { 2 } - 9 } }∫47​x2−9​dx​

Evaluate the integral in terms of natural logarithms. - ∫1e510dxx1+(ln⁡x)2\int _ { 1 } ^ { e ^ { 5 } } \frac { 10 d x } { x \sqrt { 1 + ( \ln x ) ^ { 2 } } }∫1e5​x1+(lnx)2​10dx​

Find the derivative of y. - y=sech⁡(5θ)(1−ln⁡sech⁡(5θ))y = \operatorname { sech } ( 5 \theta ) ( 1 - \ln \operatorname { sech } ( 5 \theta ) )y=sech(5θ)(1−lnsech(5θ))

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 initial value problem. - $\frac { d y } { d t } = e ^ { t } \sin \left( e ^ { t } - 7 \right) , y ( \ln 7 ) = 0$

Provide an appropriate response. -A polynomial $f ( x )$ has a degree smaller than or equal to another polynomial $g ( x )$ . Does $f = O ( g )$ and does $\mathrm { g } = \mathrm { O } ( \mathrm { f } )$ ?

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

Solve the differential equation. - $\frac { d y } { d x } = 8 x ^ { 7 } \sec y$

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

Solve the problem. -Find the length of the segment of the curve $y = \frac { 1 } { 2 } \cosh 2 x$ from $x = 0$ to $x = \ln \sqrt { 5 }$ .

Evaluate the integral. - $\int \frac { e ^ { 2 \theta } } { 1 + e ^ { 2 \theta } } d \theta$

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

Express the value of the inverse hyperbolic function in terms of natural logarithms. - $\operatorname { csch } ^ { - 1 } \left( \frac { 9 } { 4 } \right)$

Evaluate the integral in terms of an inverse hyperbolic function. - $\int _ { \sqrt { 5 } } ^ { 6 \sqrt { 5 } } \frac { d x } { \sqrt { 36 + x ^ { 2 } } }$

Evaluate the integral. - $\int \frac { \mathrm { e } ^ { 1 / x } } { 5 x ^ { 2 } } d x$

Express the value of the inverse hyperbolic function in terms of natural logarithms. - $\sinh ^ { - 1 } \left( \frac { - 3 } { 4 } \right)$

Evaluate the integral in terms of natural logarithms. - $\int _ { 4 } ^ { 7 } \frac { d x } { \sqrt { x ^ { 2 } - 9 } }$

Evaluate the integral in terms of natural logarithms. - $\int _ { 1 } ^ { e ^ { 5 } } \frac { 10 d x } { x \sqrt { 1 + ( \ln x ) ^ { 2 } } }$

Find the derivative of y. - $y = \operatorname { sech } ( 5 \theta ) ( 1 - \ln \operatorname { sech } ( 5 \theta ) )$