skyturnred said:Homework Statement
f'(x)= (2+x^2)/(1+x^2), find f(x).
Homework Equations
The Attempt at a Solution
I am pretty sure you have to use integration by parts... but I don't think we learned it yet. Is there a way to do this without using integration by parts? If not, how would I use integration by parts for his question?
gb7nash said:Well, [itex]\frac{2+x^2}{1+x^2} = \frac{2}{1+x^2}+\frac{x^2}{1+x^2}[/itex]
If you know what the integral of [itex]\frac{1}{1+x^2}[/itex] is, the first part is easy. The second part is simple as well, if you recognize what to do first before integrating.
An antiderivative is the reverse process of differentiation. It is a function that, when differentiated, will give the original function as its result.
To find the antiderivative of a function, you need to use integration techniques. For this specific function, the antiderivative can be found by using the substitution method or the power rule.
The power rule states that the antiderivative of x^n is (x^(n+1))/(n+1), where n is any real number except for -1. This can be applied to find the antiderivative of the given function.
Yes, the antiderivative of a function can have multiple solutions. This is because when we differentiate a function, we lose some information about the function, so there can be multiple functions that could have given the same derivative.
The antiderivative of f'(x)= (2+x^2)/(1+x^2) is arctan(x) + 2x. This can be verified by differentiating the answer and getting the original function as the result.