Calculus – Difference Between an Indefinite Integral and an Antiderivative

calculusterminology

I thought these were different words for the same thing, but it seems I am wrong. Help.

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"Indefinite integral" and "anti-derivative(s)" are the same thing, and are the same as "primitive(s)".

(Integrals with one or more limits "infinity" are "improper".)

Added: and, of course, usage varies. That is, it is possible to find examples of incompatible uses. And, quite seriously, $F(b)=\int_a^b f(t)\,dt$ is different from $F(x)=\int_a^x f(t)\,dt$ in what fundamental way? And from $\int_0^x f(t)\,dt$? And from the same expression when $f$ may not be as nice as we'd want?

I have no objection if people want to name these things differently, and/or insist that they are somewhat different, but I do not see them as fundamentally different.

So, the real point is just to be aware of the usage in whatever source...

(No, I'd not like to be in a classroom situation where grades hinged delicately on such supposed distinctions.)

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[Math] the difference between Riemann and Riemann-Stieltjes integrals

It seems to me that you are integrating relative to a $dg(x)$, rather than $dx$. For example, if $g(x)$ is $0$ for negative $x$ and $1$ for positive $x$, then then $\int_{-1}^{1} f(x)dg(x)$ is $f(0)$ if $f$ is continuous.

If $g(x)=x$, the Riemann-Stieltjes integral is just the Riemann integral.

If $g(x)$ is continuously differentiable, then the RS-integral $\int_{a}^{b} f(x)dg(x)$ is the same as the Riemann integral $\int_a^b f(x)g'(x) dx$.

The differences then are the cases where $g(x)$ is not continuously differentiable. For example, if $g(x)$ is the step function above, then $dg(x)$ is "like" the Dirac delta function.

It's a beginning of a way of thinking of integrals as operators.

[Math] the difference between exponentials and powers

Euler's number, $e \approx 2.71828$, is very important in many fields of mathematics. An important function that uses this constant is $\exp x = e^x$.

However, $\operatorname{pow}(a, b) = a^b$, a more general case.

You will notice that, if $a=e$,

Math.pow(Math.E, x)

is the same as

Math.exp(x)

Once you are familiar with more properties of $e^x$, you will also note that

Math.exp(b * Math.log (a))

is the same as

Math.pow(a, b)

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[Math] the difference between Riemann and Riemann-Stieltjes integrals

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