Question

one integral

Answer 1

\[\Large \int\limits e^{x^2}~dx\]

Answer 2

\[\Large e^{x}=\sum_{n=1}^{\infty}\frac{x^n}{n!}\]\[\Large e^{x^2}=\sum_{n=1}^{\infty}\frac{(x^2)^n}{n!}=\sum_{n=1}^{\infty}\frac{x^{2n}}{n!}\]\[\Large \int\limits e^{x^2}dx=~\sum_{n=1}^{\infty}~\frac{x^{2n+1}}{~(2n+1)~n!}\]

Answer 3

Do I need the +C ?

Answer 4

yes, you need the +C if you have no limits

Answer 5

limits are a process where we want to find an area under a function that is already defined for us

without limits we are just trying to determine a 'family' of function that would satisfy the results.

limits for area, no limits for +C

Answer 6

Think about it this way:

"If I added 1 to the result of the integral and then apply a derivative, would I still get the same thing?"

If yes, add C.

Answer 7

oh, so when it comes to summations we apply the same rules and logic, I see.... tnx for clarifying that for me.

Answer 8

consider this: a polynomial representation of a function is a sum of functions, integrate them one by one and we get ...
\[f' = a'+b'+c'+d'+e'+...\]
\[\int f' =\int a'+\int b'+\int c'+\int d'+\int e'+...\]
\[\int f' =(a+c_1)+(b+c_2)+(c+c_3)+(d+c_4)+(e+c_5)+...\]
the sum of the arbitrary constants is an arbitrary constant
\[\int f' =(a+b+c+d+e+...)+K\]