Question

I need help...
If f'(x) = 2x/(x^2+1), then f(x) could be...

It's a multiple choice answer but i want to know how to get the antiderivative.

Answer 1

substituion must be used to simplift the integral.
let u = x^2 + 1
du = 2x dx

this is then equivalent to du/u

integral is ln|u| + C which is ln|x^2 + 1| + C

Answer 2

simplify*

Answer 3

opposite of differentiation is called as integration
\[
f'(x)={2x\over x^2+1}\\
\implies\\
f(x)=\int {2x\over x^2+1}dx\\
\text{solve by substitution:}\quad u=x^2+1\implies du=2xdx
\]

Answer 4

alright i think i'm understanding both of you let me see what f(x) could be using substitution.

Answer 5

so pretty much like Euler said you get \[\ln |x^{2}+1| +C\] but, there are two answers that are similar but not what i got.
\[(A) 3+\ln (x ^{2}+1)\] and \[(D) x+\ln (x^2+1) +C\]

Answer 6

did they provide an initial condition?

Answer 7

no initial. pretty much that question or problem i typed up is word for word.

Answer 8

...other than the multilple choices

Answer 9

do you want me to list all of the multiple choice answers?

Answer 10

then the answer is what you have got.
C can be any real constant.

Answer 11

so A then?

Answer 12

that sounds logical

Answer 13

alright thank you.