Prove that the integral diverge.

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In summary, the conversation discusses a function f that is positive, continuous and has a domain of [0,inf). The goal is to prove that the integral of f from 0 to infinity is equal to infinity, and that the function g, defined as the integral of f from 0 to x, also has an infinite value.
  • #1
puzzek
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Let f:[0,inf) [itex]\rightarrow[/itex] R+ (f is positive...).
f is continuous and: [PLAIN]http://www3.wolframalpha.com/Calculate/MSP/MSP350919g712a61ice7c02000046b8i2i10g466041?MSPStoreType=image/gif&s=15&w=87&h=35.

prove that :[PLAIN]http://www3.wolframalpha.com/Calculate/MSP/MSP334719g7149aa8g6efha000027bhe3a6fe5cg8c9?MSPStoreType=image/gif&s=22&w=93&h=40

where [PLAIN]http://www3.wolframalpha.com/Calculate/MSP/MSP165019g717ia4c72gibe000039f8g9b652gd3664?MSPStoreType=image/gif&s=41&w=99&h=35.

some tips?
thanks.
 
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  • #2
You say that [itex]\int_0^\infty f(x) dx= \infty[/itex] and then [itex]g(x)= \int_0^\infty f(t)dt[/itex]. Did you mean [itex]g(x)= \int_0^x f(t)dt[/itex]?
 
  • #3
Yes, sorry for the mistake, this is exactly as you mentioned.
 

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

For an integral to diverge means that the value of the integral is either infinite or does not exist. This indicates that the function being integrated does not have a finite area under the curve.

2. How can you prove that an integral diverges?

To prove that an integral diverges, you can use various mathematical techniques such as the comparison test, limit comparison test, or the integral test. These methods involve comparing the given integral to a known divergent integral or evaluating the limit of the function.

3. Can an integral diverge at both the upper and lower limits?

Yes, an integral can diverge at both the upper and lower limits. This means that the function being integrated has infinite values at both ends, resulting in an infinite area under the curve.

4. What are some common types of functions that result in divergent integrals?

Some common types of functions that result in divergent integrals include power functions with exponents less than or equal to -1, logarithmic functions, and functions with discontinuities or infinite limits at one or both ends.

5. Why is it important to understand when an integral diverges?

Understanding when an integral diverges is important because it helps us determine the behavior of a given function. This knowledge can be applied in various fields such as physics and engineering, where integrals are used to calculate important quantities. It also helps us identify when a function does not have a finite area under the curve, which can have significant implications in real-life scenarios.

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