Abelian
Still Commuting
4 weeks ago#914959
we all deal with functions with nonelementary antiderivatives a lot, and they certainly are a pain to integrate. even more frustrating is when there are no special functions that have been named to solve such integrals, as has been done with things like the error function, bessel function, beta function, etc.
there is hope, though. all polynomial and rational functions have elementary antiderivatives. no matter how complicated a rational function or how high a degree polynomial, it is always integrable. polynomials are quite easy: it's just the reverse power rule. for rational functions, it is always possible through any combination of factoring, decomposition into partial fractions, and polynomial division to find your elementary solution. i hope this provides to you some comfort while tackling difficult integrals, and have a blessed day
there is hope, though. all polynomial and rational functions have elementary antiderivatives. no matter how complicated a rational function or how high a degree polynomial, it is always integrable. polynomials are quite easy: it's just the reverse power rule. for rational functions, it is always possible through any combination of factoring, decomposition into partial fractions, and polynomial division to find your elementary solution. i hope this provides to you some comfort while tackling difficult integrals, and have a blessed day
DerSoylentDrinker
KUMQUAT LEGION 2026
4 weeks ago#916300
can you dumb it down a notch?
4 weeks ago#916337
I hate when I realize someone is trying to infiltrate on here from der cord
@Vashkentya
@Vashkentya
Abelian
Still Commuting
4 weeks ago#917156
an elementary function is one that is "easy" to work with or common in some sense. pretty much any mathematical function you can think of is elementary: polynomials, rational functions, trigonometric functions, exponential functions, root functions, logarithms, and inverse trig, as well as any combination of them.can you dumb it down a notch?
a non elementary function is any function that is not listed above. they are usually harder to work with, but it is important to note that the distinction between "elementary" and "nonelementary" isn't rigorously defined, so it is just convention as to which functions are which. you are probably familiar with the absolute value function f(x) = |x|, and that function is nonelementary.
an integral is a tool in calculus used to calculate areas under curves. it is usually taught in two cases: the definite and indefinite integral. the definite integral takes a function, as well as a lower and upper bound, to calculate area. it finds the area under the curve between the two bounds provided. the indefinite integral, also sometimes called the antiderivative, is a more general case of a definite integral. it takes in a function and outputs another function. this new function is the antiderivative and it describes how the area under the original function changes when the upper bound changes.
a problem arises when trying to calculate the antiderivatives of some elementary functions, like for example sin(x)/x. first off, this function is elementary because it is just two elementary functions being divided, but trying to find its antiderivative is impossible in terms of elementary functions. this is what is meant by "nonelementary antiderivative". its antiderivative is not able to be expressed in terms of elementary functions.
rational functions (two polynomials being divided) will never have a nonelementary antiderivtives. using the methods i shared above it is always able to be expressed at worst as a mix of polynomials, logarithms, and inverse trig.
when taking the antiderivative of a polynomial, you always get another polynomial back, so they all also have elementary antiderivatives.
DerSoylentDrinker
KUMQUAT LEGION 2026
4 weeks ago#917275
too smart, can you make it a little stupider?an elementary function is one that is "easy" to work with or common in some sense. pretty much any mathematical function you can think of is elementary: polynomials, rational functions, trigonometric functions, exponential functions, root functions, logarithms, and inverse trig, as well as any combination of them.Show quoted text
can you dumb it down a notch?
a non elementary function is any function that is not listed above. they are usually harder to work with, but it is important to note that the distinction between "elementary" and "nonelementary" isn't rigorously defined, so it is just convention as to which functions are which. you are probably familiar with the absolute value function f(x) = |x|, and that function is nonelementary.
an integral is a tool in calculus used to calculate areas under curves. it is usually taught in two cases: the definite and indefinite integral. the definite integral takes a function, as well as a lower and upper bound, to calculate area. it finds the area under the curve between the two bounds provided. the indefinite integral, also sometimes called the antiderivative, is a more general case of a definite integral. it takes in a function and outputs another function. this new function is the antiderivative and it describes how the area under the original function changes when the upper bound changes.
a problem arises when trying to calculate the antiderivatives of some elementary functions, like for example sin(x)/x. first off, this function is elementary because it is just two elementary functions being divided, but trying to find its antiderivative is impossible in terms of elementary functions. this is what is meant by "nonelementary antiderivative". its antiderivative is not able to be expressed in terms of elementary functions.
rational functions (two polynomials being divided) will never have a nonelementary antiderivtives. using the methods i shared above it is always able to be expressed at worst as a mix of polynomials, logarithms, and inverse trig.
when taking the antiderivative of a polynomial, you always get another polynomial back, so they all also have elementary antiderivatives.
Abelian
Still Commuting
4 weeks ago#917310
elementary = goodtoo smart, can you make it a little stupider?Show quoted text
an elementary function is one that is "easy" to work with or common in some sense. pretty much any mathematical function you can think of is elementary: polynomials, rational functions, trigonometric functions, exponential functions, root functions, logarithms, and inverse trig, as well as any combination of them.
a non elementary function is any function that is not listed above. they are usually harder to work with, but it is important to note that the distinction between "elementary" and "nonelementary" isn't rigorously defined, so it is just convention as to which functions are which. you are probably familiar with the absolute value function f(x) = |x|, and that function is nonelementary.
an integral is a tool in calculus used to calculate areas under curves. it is usually taught in two cases: the definite and indefinite integral. the definite integral takes a function, as well as a lower and upper bound, to calculate area. it finds the area under the curve between the two bounds provided. the indefinite integral, also sometimes called the antiderivative, is a more general case of a definite integral. it takes in a function and outputs another function. this new function is the antiderivative and it describes how the area under the original function changes when the upper bound changes.
a problem arises when trying to calculate the antiderivatives of some elementary functions, like for example sin(x)/x. first off, this function is elementary because it is just two elementary functions being divided, but trying to find its antiderivative is impossible in terms of elementary functions. this is what is meant by "nonelementary antiderivative". its antiderivative is not able to be expressed in terms of elementary functions.
rational functions (two polynomials being divided) will never have a nonelementary antiderivtives. using the methods i shared above it is always able to be expressed at worst as a mix of polynomials, logarithms, and inverse trig.
when taking the antiderivative of a polynomial, you always get another polynomial back, so they all also have elementary antiderivatives.
non elementary = not good
integral = antiderivative = area
some elementary functions + integral = non elementary = not good
rational functions and polynomials + integral = always elementary = always good
DerSoylentDrinker
KUMQUAT LEGION 2026
4 weeks ago#917330
why are you using overly reddit language bro just speak normallyelementary = goodShow quoted text
too smart, can you make it a little stupider?Show quoted text
an elementary function is one that is "easy" to work with or common in some sense. pretty much any mathematical function you can think of is elementary: polynomials, rational functions, trigonometric functions, exponential functions, root functions, logarithms, and inverse trig, as well as any combination of them.
a non elementary function is any function that is not listed above. they are usually harder to work with, but it is important to note that the distinction between "elementary" and "nonelementary" isn't rigorously defined, so it is just convention as to which functions are which. you are probably familiar with the absolute value function f(x) = |x|, and that function is nonelementary.
an integral is a tool in calculus used to calculate areas under curves. it is usually taught in two cases: the definite and indefinite integral. the definite integral takes a function, as well as a lower and upper bound, to calculate area. it finds the area under the curve between the two bounds provided. the indefinite integral, also sometimes called the antiderivative, is a more general case of a definite integral. it takes in a function and outputs another function. this new function is the antiderivative and it describes how the area under the original function changes when the upper bound changes.
a problem arises when trying to calculate the antiderivatives of some elementary functions, like for example sin(x)/x. first off, this function is elementary because it is just two elementary functions being divided, but trying to find its antiderivative is impossible in terms of elementary functions. this is what is meant by "nonelementary antiderivative". its antiderivative is not able to be expressed in terms of elementary functions.
rational functions (two polynomials being divided) will never have a nonelementary antiderivtives. using the methods i shared above it is always able to be expressed at worst as a mix of polynomials, logarithms, and inverse trig.
when taking the antiderivative of a polynomial, you always get another polynomial back, so they all also have elementary antiderivatives.
non elementary = not good
integral = antiderivative = area
some elementary functions + integral = non elementary = not good
rational functions and polynomials + integral = always elementary = always good
Abelian
Still Commuting
4 weeks ago#917352
sorry i thought you were being tongue in cheek about itwhy are you using overly reddit language bro just speak normallyShow quoted text
elementary = good
non elementary = not good
integral = antiderivative = area
some elementary functions + integral = non elementary = not good
rational functions and polynomials + integral = always elementary = always good
Log in to reply to this thread.