How to integrate using first principles ?

Question

anonymous · Answer

Integrate using first principles...?
Please elaborate.

kropot72 · Answer

The general rule for integration is$$\int\limits_{}^{}ax ^{n} dx=\frac{a}{n+1}x ^{n+1}+c$$
where n is not equal to -1

anonymous · Answer

And, of course, if $n=-1$:
$$\int ax^{-1}dx=a\ln (x)+C$$

unklerhaukus · Answer

$$\int e^{ax}\text dx=\frac {e^{ax}}a+c$$

amistre64 · Answer

first principles deal with taking the limit of an algebraic manipulation

amistre64 · Answer

$$\lim_{h\to\ 0}\frac{f(x+h)-f(x)}{h}$$

amistre64 · Answer

thats derivative tho lol

amistre64 · Answer

integration is same concept, but with summations

amistre64 · Answer

$$\lim_{n\to\ inf}\ \sum_{i=1}^{n}\frac{b-a}{n}f(a+\frac{b-a}{n}i)$$

or something like that

anonymous · Answer

I think you are looking for Riemannian sums, Amistre. That is the outdated view of integration. (I do not know how outdated it is, but that seems to be the majority opinion.)

amistre64 · Answer

reimanns, yes.  that was my thought

unklerhaukus · Answer

Is this a question abou the fundamental theorem of calculus ?

anonymous · Answer

Well, I think it is a rather productive discussion of basic calculus right now.

amistre64 · Answer

... or as productive as it can be at 8 in the morning ;)

anonymous · Answer

True...


I was going to explain the Riemann sum, but there are way too many details for it to be interesting right now... -_-