This article is about a particular function from a subset of the real numbers to the real numbers. Information about the function, including its domain, range, and key data relating to graphing, differentiation, and integration, is presented in the article.
View a complete list of particular functions on this wiki
For functions involving angles (trigonometric functions, inverse trigonometric functions, etc.) we follow the convention that all angles are measured in radians. Thus, for instance, the angle of is measured as .
is written as for shorthand.
|Default domain||all real numbers except the odd integer multiples of .|
|range|| , i.e., . All nonnegative reals.|
no absolute maximum value; absolute minimum value of 0
|period||, i.e., .|
|local maximum values and points of attainment||No local maximum values|
|local minimum values and points of attainment||0 at all integer multiples of .|
|points of inflection (both coordinates)||None|
|vertical asymptotes||at all odd multiples of , with the function going to from both directions in each case.|
|antiderivative||. Note that the value must be constant within each interval between consecutive odd multiples of , but can be different in different intervals. The domain overall is not connected.|
|interval description based on increase/decrease and concave up/down|| For each integer :|
decreasing and concave up from to
increasing and concave up from to .
We use the identity:
Using this, we rewrite:
We can antidifferentiate the function one more time:
Integration of products with polynomials
Using integration by parts, we know that if it is possible to integrate a function times, it is also possible to use that information to integrate times the function. Thus, from the above, we can integrate within elementary functions. However, it is not possible to integrate within elementary functions.
Explicitly, we have: