Absolute values in standard integrals

In summary, there are a lot of standard integrals that involve trigonometric functions with an absolute value in their solution. This may seem out of place, as the integral of dx \cot{x} equals \log |\sin{x}|, but it could be mistaken to just use \log (\sin{x}). It is possible that the use of | | is not for their meaning as absolute value, and there is no point in using them instead of regular brackets. Furthermore, the solution may not be smooth due to the unbounded nature of the cotangent function.
  • #1
NanakiXIII
392
0
In a lot of compilations of standard integrals (my Calculus book does this, Wikipedia does this), a lot of the integrals of trigonometric functions have an absolute value in their solution which seems out of place to me. For example, take the integral

[tex]\int dx \cot{x}[/tex].

My Calculus book says this equals [tex]\log |\sin{x}|[/tex]. Now, I could be mistaken, but it seems to me that [tex]\log (\sin{x})[/tex] would do just fine and in fact, since [tex]|\sin{x}|[/tex] isn't a smooth function, I wouldn't expect to find it as a solution.

My first guess is that the [tex]| |[/tex] are for some reason used without their meaning as absolute value. Is this the case? If so, what's the point of using them instead of regular brackets?
 
Physics news on Phys.org
  • #2
log(-x) isn't defined.
 
  • #3
You are right, of course. And the cotangent is not a bounded function, so the solution not being smooth is not so strange.
 

Related to Absolute values in standard integrals

1. What is an absolute value in a standard integral?

An absolute value in a standard integral refers to the distance of a number from 0 on a number line. It is represented by the symbol "|" and can be positive or negative.

2. Why are absolute values important in standard integrals?

Absolute values are important in standard integrals because they help to ensure that the integral is always positive, even if the function being integrated has negative values. This allows for more accurate calculations and helps to avoid errors.

3. How do you find the absolute value in a standard integral?

To find the absolute value in a standard integral, you simply take the absolute value of the integrand. This means that you ignore any negative signs and only consider the magnitude of the number.

4. Can absolute values be used in all types of integrals?

Yes, absolute values can be used in all types of integrals, including definite and indefinite integrals. They are particularly useful in improper integrals, where the function being integrated may have negative values or infinite limits.

5. Are there any special rules for working with absolute values in standard integrals?

One special rule for working with absolute values in standard integrals is the absolute value property, which states that the absolute value of the integral of a function is equal to the integral of the absolute value of that function. This can be useful in certain calculations and simplifications.

Similar threads

B Integral of cosecant function: understanding different approaches
    • Calculus
Replies
14
Views
1K
I Why is the absolute value of sinx used in the integral of cotx?
    • Calculus
Replies
1
Views
787
I Did Math DF's integral calculator make a glaring mistake?
    • Calculus
Replies
8
Views
406
B Are Both Answers Correct for Trigonometric Substitution Integral?
    • Calculus
Replies
5
Views
1K
B Trigonometric substitution, a case I'd like to share
    • Calculus
Replies
3
Views
1K
A Multiplying divergent integrals using Hardy fields approach
    • Calculus
Replies
3
Views
1K
I Why is the integral of ##\arcsin(\sin(x))## so divisive?
    • Calculus
Replies
3
Views
1K
I Substitution in a definite integral
    • Calculus
Replies
6
Views
1K
I Why Does Integrating |f(x)| Differ from Integrating f(x)?
    • Calculus
Replies
5
Views
1K
I How to evaluate the enclosed area of this implicit curve?
    • Calculus
Replies
2
Views
2K

Hot Threads

Recent Insights

Back
Top