Is the sign of the integral of this function negative?

In summary, the conversation discusses a function f(x) that is continuous and differentiable on the interval [0,1), with a limit of positive infinity as x approaches 1 from the left. The function also has a finite integral A on the interval, and the question asks whether it is possible for the sign of A to be negative. An example is provided where f(x) is negative for much of the interval, implying that it is possible for the sign of A to be negative.
  • #1
Rlwe
18
1
Let ##f:[0;1)\to\mathbb{R}## and ##f\in C^1([0;1))## and ##\lim_{x\to1^-}f(x)=+\infty## and ##\forall_{x\in[0;1)}-\infty<f(x)<+\infty##. Define $$A:=\int_0^1f(x)\, dx\,.$$ Assuming ##A## exists and is finite, is it possible that ##\text{sgn}(A)=-1##?
 
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  • #2
f(x) < 0 for much of the interval.
 
  • #3
Rlwe said:
Let ##f:[0;1)\to\mathbb{R}## and ##f\in C^1([0;1))## and ##\lim_{x\to1^-}f(x)=+\infty## and ##\forall_{x\in[0;1)}-\infty<f(x)<+\infty##. Define $$A:=\int_0^1f(x)\, dx\,.$$ Assuming ##A## exists and is finite, is it possible that ##\text{sgn}(A)=-1##?
Try $$f(x)=\frac{1}{\sqrt{1-x}}-3.$$
 

Related to Is the sign of the integral of this function negative?

1. What is an improper integral?

An improper integral is an integral where one or both of the limits of integration are infinite or the function being integrated has a singularity within the interval of integration. This means that the integral does not have a finite value and requires special techniques to evaluate.

2. How do you determine if an integral is improper?

To determine if an integral is improper, you must check if any of the following conditions are met: the lower limit of integration is infinite, the upper limit of integration is infinite, or there is a singularity within the interval of integration. If any of these conditions are met, then the integral is considered improper.

3. How do you evaluate an improper integral?

To evaluate an improper integral, you must first determine the type of improper integral it is (infinite limit, singularity, or both). Then, you can use various techniques such as limits, comparison, or substitution to evaluate the integral. In some cases, the improper integral may not have a finite value.

4. What is the significance of an improper integral?

Improper integrals are important in mathematics and science because they allow us to integrate functions that would otherwise be impossible to integrate. They also have applications in physics, engineering, and other fields where infinite or singular values may arise.

5. Are there any restrictions when using improper integrals?

Yes, there are some restrictions when using improper integrals. For example, the function being integrated must be continuous within the interval of integration, except for a possible singularity. Also, the integral must converge (have a finite value) in order for it to be evaluated. If the integral does not meet these restrictions, it cannot be evaluated using traditional methods.

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