How do you determine if the improper integral converges or diverges #int dx/((3x-2)^6) # from 2 to infinity?

Answer 1

The integral is convergent and:

#int_2^oo (dx)/(3x-2)^6 = 1/15360#

By definition:

#int_2^oo (dx)/(3x-2)^6 = lim_(t->oo) int_2^t (dx)/(3x-2)^6 = lim_(t->oo) -1/15 [1/(3x-2)^5]_2^t#

We have that:

#lim_(t->oo) 1/(3t-2)^5 = 0#

So:

#int_2^oo (dx)/(3x-2)^6 = 1/15 1/4^5 = 1/15360#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To determine if the improper integral ( \int_{2}^{\infty} \frac{1}{(3x - 2)^6} , dx ) converges or diverges, we need to analyze its behavior as ( x ) approaches infinity.

We'll use the limit comparison test with a known integral. Let's compare it to the integral ( \int_{2}^{\infty} \frac{1}{x^6} , dx ).

[ \lim_{x \to \infty} \frac{\frac{1}{(3x - 2)^6}}{\frac{1}{x^6}} ]

By simplifying, we get:

[ \lim_{x \to \infty} \frac{x^6}{(3x - 2)^6} ]

[ = \lim_{x \to \infty} \left(\frac{x}{3x - 2}\right)^6 ]

As ( x ) approaches infinity, the expression ( \frac{x}{3x - 2} ) approaches ( \frac{1}{3} ).

Thus, the limit becomes ( \left(\frac{1}{3}\right)^6 = \frac{1}{729} ).

Since the limit is a finite non-zero number, and the integral ( \int_{2}^{\infty} \frac{1}{x^6} , dx ) converges (it's a p-series with ( p > 1 )), by the limit comparison test, the given integral ( \int_{2}^{\infty} \frac{1}{(3x - 2)^6} , dx ) also converges.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7