Find the values of p for which the integral converges?

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In summary, the problem asks to find the values of p for which the integral ∫ 0->1 1/(x^p) dx converges and then evaluate the integral for those values of p. After some testing, the answer in the back of the testbook suggests that the integral converges for p<1 and the value of the integral is 1/(1-p).
  • #1
Kaede_N9
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Homework Statement


Find the values of p for which the integral converges and evaluate the integral for those values of p.

∫ 0->1 1/(x^p) dx

Homework Equations


None.


The Attempt at a Solution



First thought:

Since we must evaluate 0 to 1, 1/0 is undefined so maybe 1/ (0^0) = 1.
I don't think this is correct.

Second thought:
If the first thought didnt work, how about lim x->0.
Test:
p≥1
1/ (0.000000000000000000...01)^1
≈ ∞

0>p>1
1/ (0.000000000000000000...01)^.5
≈ ∞

p<0
1/ (0.000000000000000000...01)^-1
≈ 0

If this test is true, I am not sure how to evaluate for 1.

4. Answer in the back of the testbook
p<1 , 1/(1-p)

From the answer, I am not sure how p<1 would work (but I do understand p<0).
 
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  • #2
You are making a test to guess if f(x)=1/x^p diverges at x=0. That doesn't tell you the integral necessarily diverges. Take your p=1/2 case. What the antiderivative of 1/x^(1/2)?
 

FAQ: Find the values of p for which the integral converges?

1. What does it mean for an integral to converge?

When an integral converges, it means that the value of the integral approaches a finite number as the limits of integration get closer and closer together. In other words, the area under the curve is finite and can be calculated accurately.

2. How can you tell if an integral converges or diverges?

To determine if an integral converges or diverges, you can use a variety of techniques such as the comparison test, the limit comparison test, or the ratio test. These tests involve evaluating the behavior of the integrand as the limits of integration approach infinity.

3. What are the common values of p for which the integral converges?

The most commonly encountered values of p for which the integral converges are between 1 and 2. This includes values such as p = 1, p = 1/2, and p = 2. However, there are many more values of p for which the integral can converge, depending on the specific integrand.

4. How does the value of p affect the convergence of an integral?

The value of p has a significant impact on the convergence of an integral. In general, as p increases, the integral is more likely to converge. This is because larger values of p make the integrand decrease more quickly, resulting in a smaller area under the curve. However, there are exceptions to this rule, and the specific behavior of the integrand must also be considered.

5. Can an integral converge for values of p outside the range of 1 to 2?

Yes, an integral can converge for values of p outside the range of 1 to 2. Some integrals may converge for values of p less than 1, such as p = 1/4, while others may converge for values of p greater than 2, such as p = 3. It is important to use the appropriate convergence tests to determine the convergence of the integral for a specific value of p.

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