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svetlana [45]
3 years ago
7

Determine the interval on which f(x) = ln(x) is integrable.

Mathematics
2 answers:
Juliette [100K]3 years ago
8 0
<span>a. (0, ∞)
....................................</span>
PilotLPTM [1.2K]3 years ago
3 0
<h2>Answer:</h2>

  Option: a is the correct answer.

                  a.      (0, ∞)

<h2>Step-by-step explanation:</h2>

We are given a logarithmic function f(x) as:

f(x)=\ln x

We know that the logarithmic function is defined for all the real values strictly greater than 0 i.e. x>0.

i.e. the function is defined for all positive real numbers.

i.e. the domain of the function f(x) is: (0,∞).

Also, we know that the function f(x) is integrable in it's domain and the integration is calculated by using the integration by parts.

i.e.

\int\limits {\ln x} \, dx=\int\limits {1\cdot \ln x} \, dx\\  \\i.e.\\\\\int\limits {\ln x} \, dx=\ln x\cdot \int\limits {1} \, dx-\int\limits {\dfrac{d}{dx}\ln x} \cdot \int\limits {1} \, dx\\\\i.e.\\\\\int\limits {\ln x} \, dx=\ln x\cdot x-\int\limits {\dfrac{1}{x}\cdot x} \, dx\\\\i.e.\\\\\int\limits {\ln x} \, dx=x\cdot \ln x-\int\limits {1} \, dx\\\\i.e.\\\\\int\limits {\ln x} \, dx=x\cdot \ln x-x

Hence, the answer is: Option: a

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Elis [28]

Answer:

x = 36

Step-by-step explanation:

x - 12\sqrt{x} + 36 = 0

Subtract x and 36 from both sides.

-12\sqrt{x} = -x - 36

Divide both sides by -1.

12\sqrt{x} = x + 36

Square both sides.

144x = x^2 + 72x + 1296

Subtract 144x from both sides.

0 = x^2 - 72x + 1296

Factor the right side.

0 = (x - 36)^2

x - 36 = 0

x = 36

Since the solution of the equation involved squaring both sides, we musty check the answer for possible extraneous solutions.

Check x = 36:

x - 12\sqrt{x} + 36 = 0

36 - 12\sqrt{36} + 36 = 0

36 - 12\times 6 + 36 = 0

36 - 72 + 36 = 0

0 = 0

Since 0 = 0 is a true statement, the solution x = 36 is a valid solution.

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3 years ago
A ski resort has 3 lifts, each with access to 6 ski trails. Explain how you can find the number of possible outcomes when choosi
svetlana [45]
You could multiply 6 times 3 to get your answer. The answer would be 18

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Step-by-step explanation:


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Illusion [34]

Given:

Total number of students = 27

Students who play basketball = 7

Student who play baseball = 18

Students who play neither sports = 7

To find:

The probability the student chosen at randomly from the class plays both basketball and base ball.

Solution:

Let the following events,

A : Student plays basketball

B : Student plays baseball

U : Union set or all students.

Then according to given information,

n(U)=27

n(A)=7

n(B)=18

n(A'\cap B')=7

We know that,

n(A\cup B)=n(U)-n(A'\cap B')

n(A\cup B)=27-7

n(A\cup B)=20

Now,

n(A\cup B)=n(A)+n(B)-n(A\cap B)

20=7+18-n(A\cap B)

n(A\cap B)=7+18-20

n(A\cap B)=25-20

n(A\cap B)=5

It means, the number of students who play both sports is 5.

The probability the student chosen at randomly from the class plays both basketball and base ball is

\text{Probability}=\dfrac{\text{Number of students who play both sports}}{\text{Total number of students}}

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Therefore, the required probability is \dfrac{5}{27}.

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