1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
adoni [48]
3 years ago
14

Integrate the following

Mathematics
1 answer:
enot [183]3 years ago
7 0

I suppose you mean to have the entire numerator under the square root?

\displaystyle\int_2^4\frac{\sqrt{x^2-4}}{x^2}\,\mathrm dx

We can use a trigonometric substitution to start:

x=2\sec t\implies\mathrm dx=2\sec t\tan t\,\mathrm dt

Then for x=2, t=\sec^{-1}1=0; for x=4, t=\sec^{-1}2=\frac\pi3. So the integral is equivalent to

\displaystyle\int_0^{\pi/3}\frac{\sqrt{(2\sec t)^2-4}}{(2\sec t)^2}(2\sec t\tan t)\,\mathrm dt=\int_0^{\pi/3}\frac{\tan^2t}{\sec t}\,\mathrm dt

We can write

\dfrac{\tan^2t}{\sec t}=\dfrac{\frac{\sin^2t}{\cos^2t}}{\frac1{\cos t}}=\dfrac{\sin^2t}{\cos t}=\dfrac{1-\cos^2t}{\cos t}=\sec t-\cos t

so the integral becomes

\displaystyle\int_0^{\pi/3}(\sec t-\cos t)\,\mathrm dt=\boxed{\ln(2+\sqrt3)-\frac{\sqrt3}2}

You might be interested in
A rectangular prism has a volume of 729ft^3. The length width and height are the same. What is the length of each side
viktelen [127]

Answer:

9 feet

Step-by-step explanation:

If each side is the equivalent, each side of the prism is the cube root of 729

Cube root of 729 is 9

Each side is 9 feet long

3 0
3 years ago
The basketball boosters are selling school t-shirts for 20 dollars and pants for 45 dollars. Last week they sold 100 items and c
lana66690 [7]

He sold 76 shirts and 24 pants.

Step-by-step explanation:

Given,

Cost of one t-shirt = $20

Cost of one pants = $45

Total items sold = 100

Total sales = 2600

Let,

x be the number of t-shirts

y be the number of pants

According to given statement;

x+y=100    Eqn 1

20x+45y=2600   Eqn 2

Multiplying Eqn 1 by 20

20(x+y=100)\\20x+20y=2000\ \ \ Eqn\ 3

Subtracting Eqn 3 from Eqn 2

(20x+45y)-(20x+20y)=2600-2000\\20x+45y-20x-20y=600\\25y=600

Dividing both sides by 25

\frac{25y}{25}=\frac{600}{25}\\y=24

Putting in Eqn 1

x+24=100\\x=100-24\\x=76

He sold 76 shirts and 24 pants.

Keywords: linear equations, subtraction

Learn more about linear equations at:

  • brainly.com/question/289820
  • brainly.com/question/2826639

#LearnwithBrainly

3 0
3 years ago
Xavier is testing different types of planting soil to determine which type is most effective in growing blueberry bushes. He pur
vladimir1956 [14]

Answer:

He did not control for lurking variables and their impacts on the results of his experiment. Amount of sunlight and water received are two outside variables(or confounding variables) that may impact the growth of his plants and influence the results. He needs to apply the same amount of sunlight and water to each plant within a different planting soil in order to rule out the influence of those two variables and test the sole effect of the soil brand on the plant growth. Otherwise, it would be hard to determine whether his plant growth was because of the soil brand or the different amounts of sunlight and water received

4 0
2 years ago
Five less than the product of 14 and Vanessa's height Use the variable v to represent Vanessa's height.
Fudgin [204]

Answer:

14v - 5

Step-by-step explanation:

The product of 14 and v is 14v. 5 less than that is 14v - 5.

7 0
3 years ago
!; What would the ordered pair be?
Evgen [1.6K]

Answer:

(3, 2)

Step-by-step explanation:

y = 3x - 7

2x + 5y = 16

Solve for x:

2x + 5(3x - 7) = 16

2x + 15x - 35 = 16

17x = 51

x = 3

Solve for y:

y = 3(3) - 7

y = 9 - 7

y = 2

Answer:

(3, 2)

7 0
2 years ago
Read 2 more answers
Other questions:
  • Sara is building a circular garden with a radius of 14 m. She can buy the fencing in
    10·2 answers
  • A number cube is rolled and a spinner is spun. Find the probability. The spinner has five sections. 3 sections have the letter B
    11·1 answer
  • What is the sum of -7 and its additive inverse?
    8·2 answers
  • 14389+4309 what is it
    13·2 answers
  • When solving an equation, Carmen's first step is shown below. Which property justifies Carmen's first step? plz help me
    5·1 answer
  • Need help with these
    9·1 answer
  • Find the area of the parallelogram.
    9·1 answer
  • Round 51.78 to the greatest number.
    10·2 answers
  • Two cards are drawn from a pack of 52 cards one by one without replacement; that is, the first card is not replaced before the s
    10·1 answer
  • What is the value of x? (Serious answers only plz)
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!