Calculus with the inverse hyperbolic functions

Calculus with the inverse hyperbolic functions

12th Grade - University

9 Qs

quiz-placeholder

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Calculus with the inverse hyperbolic functions

Calculus with the inverse hyperbolic functions

Assessment

Quiz

Mathematics

12th Grade - University

Hard

Created by

Oxana OLAROU

Used 23+ times

FREE Resource

9 questions

Show all answers

1.

MULTIPLE SELECT QUESTION

1 min • 1 pt

Which statements below are correct?


More than 1 correct answer is possible.

ddx(arsinhx)=1x21\frac{d}{dx}\left(\operatorname{arsinh}x\right)=\frac{1}{\sqrt{x^2-1}}

ddx(arsinhx)=11+x2\frac{d}{dx}\left(\operatorname{arsinh}x\right)=\frac{1}{\sqrt{1+x^2}}

ddx(arcoshx)=1x21\frac{d}{dx}\left(\operatorname{arcosh}x\right)=\frac{1}{\sqrt{x^2-1}}

ddx(artanhx)=11x2\frac{d}{dx}\left(\operatorname{artanh}x\right)=\frac{1}{1-x^2}

2.

MULTIPLE SELECT QUESTION

1 min • 1 pt

Which statements below are correct?


More than 1 correct answer is possible.

(1x2a2)dx=arcosh(xa)+C, x>a\int_{ }^{ }\left(\frac{1}{\sqrt{x^2-a^2}}\right)dx=\operatorname{arcosh}\left(\frac{x}{a}\right)+C,\ x>a

1a2x2dx=1aartanh(xa)+C, x<a\int_{ }^{ }\frac{1}{a^2-x^2}dx=\frac{1}{a}\operatorname{artanh}\left(\frac{x}{a}\right)+C,\ \left|x\right|<a

(1x2+a2)dx=1aarsinh(xa)+C\int_{ }^{ }\left(\frac{1}{\sqrt{x^2+a^2}}\right)dx=\frac{1}{a}\operatorname{arsinh}\left(\frac{x}{a}\right)+C

(1x2+a2)dx=arsinh(xa)+C\int_{ }^{ }\left(\frac{1}{\sqrt{x^2+a^2}}\right)dx=\operatorname{arsinh}\left(\frac{x}{a}\right)+C

3.

MULTIPLE CHOICE QUESTION

1 min • 1 pt

The derivative of

 arcosh2x\operatorname{arcosh}2x  
is

 24x21\frac{2}{\sqrt{4x^2-1}}  

 14x21\frac{1}{\sqrt{4x^2-1}}  

 2x21\frac{2}{\sqrt{x^2-1}}  

 1x21\frac{1}{\sqrt{x^2-1}}  

4.

MULTIPLE CHOICE QUESTION

1 min • 1 pt

The derivative of

 artanh(1x)\operatorname{artanh}\left(\frac{1}{x}\right)  
is

 11x2\frac{1}{1-x^2}  

 1x21\frac{1}{x^2-1}  

 1x2(1x2)\frac{1}{x^2\left(1-x^2\right)}  

 x2x21\frac{x^2}{x^2-1}  

5.

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

The integral of

 116+25x2dx\int_{ }^{ }\frac{1}{\sqrt{16+25x^2}}dx  
is

 14arsinh(5x4)+C\frac{1}{4}\operatorname{arsinh}\left(\frac{5x}{4}\right)+C  

 15arsinh(5x4)+C\frac{1}{5}\operatorname{arsinh}\left(\frac{5x}{4}\right)+C  

 14arcosh(5x4)+C\frac{1}{4}\operatorname{arcosh}\left(\frac{5x}{4}\right)+C  

 15arsinh(5x4)+C\frac{1}{5}\operatorname{arsinh}\left(\frac{5x}{4}\right)+C  

6.

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

The value of 
 1.53116x29dx\int_{1.5}^3\frac{1}{\sqrt{16x^2-9}}dx  
is

 ln(4+152+3)\ln\left(\frac{4+\sqrt{15}}{2+\sqrt{3}}\right)  

 14ln(41523)\frac{1}{4}\ln\left(\frac{4-\sqrt{15}}{2-\sqrt{3}}\right)  

 14ln(4+152+3)\frac{1}{4}\ln\left(\frac{4+\sqrt{15}}{2+\sqrt{3}}\right)  

 ln(41523)\ln\left(\frac{4-\sqrt{15}}{2-\sqrt{3}}\right)  

7.

MULTIPLE CHOICE QUESTION

3 mins • 1 pt

The integral 19+8x+2x2dx\int_{ }^{ }\frac{1}{\sqrt{9+8x+2x^2}}dx  

is

 12arsinh2(x+2)+C\frac{1}{\sqrt{2}}\operatorname{arsinh}\sqrt{2}\left(x+2\right)+C  

 12arcosh2(x+2)+C\frac{1}{\sqrt{2}}\operatorname{arcosh}\sqrt{2}\left(x+2\right)+C  

 12arsinh2(x+2)+C\frac{1}{2}\operatorname{arsinh}2\left(x+2\right)+C  

 12arcosh2(x+2)+C\frac{1}{2}\operatorname{arcosh}2\left(x+2\right)+C  

8.

MULTIPLE CHOICE QUESTION

3 mins • 1 pt

The integral 13x212x4dx\int_{ }^{ }\frac{1}{\sqrt{3x^2-12x-4}}dx  

is

 arcosh(3(x2)4)+C\operatorname{arcosh}\left(\frac{\sqrt{3}\left(x-2\right)}{4}\right)+C  

 13arcosh(3(x2)4)+C\frac{1}{\sqrt{3}}\operatorname{arcosh}\left(\frac{\sqrt{3}\left(x-2\right)}{4}\right)+C  

 arsinh(3(x2)4)+C\operatorname{arsinh}\left(\frac{\sqrt{3}\left(x-2\right)}{4}\right)+C  

 13arsinh(3(x2)4)+C\frac{1}{\sqrt{3}}\operatorname{arsinh}\left(\frac{\sqrt{3}\left(x-2\right)}{4}\right)+C  

9.

MULTIPLE CHOICE QUESTION

3 mins • 1 pt

The integral 1712x4x2dx\int_{ }^{ }\frac{1}{7-12x-4x^2}dx  

is

 14artanh(2x+32)+C\frac{1}{4}\operatorname{artanh}\left(\frac{2x+3}{2}\right)+C  

 14artanh(x+32)+C\frac{1}{4}\operatorname{artanh}\left(\frac{x+3}{2}\right)+C  

 18artanh(x+34)+C\frac{1}{8}\operatorname{artanh}\left(\frac{x+3}{4}\right)+C  

 18artanh(2x+34)+C\frac{1}{8}\operatorname{artanh}\left(\frac{2x+3}{4}\right)+C