Question

What would be the integral of i^x according to you people?

Answer 1

The integral? Do you mean the anti-derivative?

Answer 2

It's weird because in some cases it gets to have more than one value like \[\sqrt{i^4}=i ^{4/2}=i^2\] but also\[\sqrt{i^4}=\sqrt{i*i*i*i}=\sqrt1\]

Answer 3

yup, say..... the area, the definite integral

Answer 4

forgot to mention, in that case then \[\sqrt{i^4}= i^2=-1\]

Answer 5

and also \[\sqrt{i^4}= \sqrt1=1\]

Answer 6

you people?

Answer 7

and in order to determine tha area of the function you would need to take both values into consideration plus the imaginary part.
What I think is that the answer would be \[2x+ix\] or something similar, I'm not sure, just like a hunch or a possibility maybe

Answer 8

where are we taking the area from?

Answer 9

1) sqrt(i^4)!=sqrt(i^2)*sqrt(i^2), that property of exponents only holds for positive numbers, sqrt(-1*-1)!=sqrt(-1)sqrt(-1)
2) to take the integral, start by rewriting i^x as e^(log(i)x)

Answer 10

the definite integral is the infinitesimal sum of infinitesimally small parts of the function, it expresses the area below the line in relation to the axis x (in case you are doing it with respect of x of course)
And now that I think about it, we have an infinite number of fractional values for X and if all of them give me 2 possible values 1 and -1, then it'd be 0

Answer 11

ohh really? well that's really good to know, I didn't know that

Answer 12

integral(e^(log(i)x)dx) = e^(log(i)x)/log(i)

Answer 13

of course its funny to make a log that has base i to any real number :P

Answer 14

hilarious!

Answer 15

there is no possible value for it then?

Answer 16

to simply, use the fact that e^(ix)=cos(x)+isin(x) (euler's formula) and plug in for x=pi/2:
e^(pi i/2)=i
thus pi i/2=log(i)

Answer 17

e^(log(i)x)/log(i) = i^x / log(i) = i^x / (pi i/2) = -2i*(i^x)/pi
thus the antiderivative is -2i*(i^x)/pi and the integral from x=a to x=b is equal to 2i*(i^a-i^b)/pi

Answer 18

awesome! pi=3.141592.... or are we talking about somthing else in complex nubers?

Answer 19

yup, that's the value of pi :)

Answer 20

awesome, I love you hahaha