UPSC Exam > UPSC Questions > The value of the improper integral integratio...
Start Learning for Free
The value of the improper integral integration 1 tend to infinity 1/x^5dx is given by A. 1/4. B. 1/6. C. 6. D. -5/6.?
Most Upvoted Answer
The value of the improper integral integration 1 tend to infinity 1/x^...
Calculation of Improper Integral
To find the value of the improper integral ∫(1 to infinity) 1/x^5 dx, we need to determine if the integral converges or diverges.
Convergence Test
We can use the p-test for convergence, which states that ∫(1 to infinity) 1/x^p dx converges if p > 1 and diverges if p ≤ 1.
In this case, p = 5, so the integral converges.
Calculating the Integral
To calculate the integral, we need to find the antiderivative of 1/x^5, which is -1/(4x^4).
Next, we evaluate the integral from 1 to t as t tends to infinity:
∫(1 to t) 1/x^5 dx = [-1/(4x^4)] from 1 to t
= [-1/(4t^4)] - [-1/(4(1)^4)]
= -1/(4t^4) + 1/4
Limit as t tends to infinity
As t tends to infinity, the value of the integral becomes:
lim(t→∞) [-1/(4t^4) + 1/4] = 0 + 1/4 = 1/4
Therefore, the value of the improper integral ∫(1 to infinity) 1/x^5 dx is A. 1/4.
To find the value of the improper integral ∫(1 to infinity) 1/x^5 dx, we need to determine if the integral converges or diverges.
Convergence Test
We can use the p-test for convergence, which states that ∫(1 to infinity) 1/x^p dx converges if p > 1 and diverges if p ≤ 1.
In this case, p = 5, so the integral converges.
Calculating the Integral
To calculate the integral, we need to find the antiderivative of 1/x^5, which is -1/(4x^4).
Next, we evaluate the integral from 1 to t as t tends to infinity:
∫(1 to t) 1/x^5 dx = [-1/(4x^4)] from 1 to t
= [-1/(4t^4)] - [-1/(4(1)^4)]
= -1/(4t^4) + 1/4
Limit as t tends to infinity
As t tends to infinity, the value of the integral becomes:
lim(t→∞) [-1/(4t^4) + 1/4] = 0 + 1/4 = 1/4
Therefore, the value of the improper integral ∫(1 to infinity) 1/x^5 dx is A. 1/4.
Attention UPSC Students!
To make sure you are not studying endlessly, EduRev has designed UPSC study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in UPSC.
|
Explore Courses for UPSC exam
|
|
Top Courses for UPSCView all
The value of the improper integral integration 1 tend to infinity 1/x^5dx is given by A. 1/4. B. 1/6. C. 6. D. -5/6.?
Question Description
The value of the improper integral integration 1 tend to infinity 1/x^5dx is given by A. 1/4. B. 1/6. C. 6. D. -5/6.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The value of the improper integral integration 1 tend to infinity 1/x^5dx is given by A. 1/4. B. 1/6. C. 6. D. -5/6.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The value of the improper integral integration 1 tend to infinity 1/x^5dx is given by A. 1/4. B. 1/6. C. 6. D. -5/6.?.
The value of the improper integral integration 1 tend to infinity 1/x^5dx is given by A. 1/4. B. 1/6. C. 6. D. -5/6.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The value of the improper integral integration 1 tend to infinity 1/x^5dx is given by A. 1/4. B. 1/6. C. 6. D. -5/6.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The value of the improper integral integration 1 tend to infinity 1/x^5dx is given by A. 1/4. B. 1/6. C. 6. D. -5/6.?.
Solutions for The value of the improper integral integration 1 tend to infinity 1/x^5dx is given by A. 1/4. B. 1/6. C. 6. D. -5/6.? in English & in Hindi are available as part of our courses for UPSC.
Download more important topics, notes, lectures and mock test series for UPSC Exam by signing up for free.
Here you can find the meaning of The value of the improper integral integration 1 tend to infinity 1/x^5dx is given by A. 1/4. B. 1/6. C. 6. D. -5/6.? defined & explained in the simplest way possible. Besides giving the explanation of
The value of the improper integral integration 1 tend to infinity 1/x^5dx is given by A. 1/4. B. 1/6. C. 6. D. -5/6.?, a detailed solution for The value of the improper integral integration 1 tend to infinity 1/x^5dx is given by A. 1/4. B. 1/6. C. 6. D. -5/6.? has been provided alongside types of The value of the improper integral integration 1 tend to infinity 1/x^5dx is given by A. 1/4. B. 1/6. C. 6. D. -5/6.? theory, EduRev gives you an
ample number of questions to practice The value of the improper integral integration 1 tend to infinity 1/x^5dx is given by A. 1/4. B. 1/6. C. 6. D. -5/6.? tests, examples and also practice UPSC tests.
|
|
Explore Courses for UPSC exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup