Given y'=e^(arctan x)/(x^2+1), what is the function y?
To determine the primitive function, we'll have to calculate the indefinite integral of y'.
We'll apply substitution technique, replacing arctan x by t.
arctan x = t
We'll differentiate both sides and we'll get:
dx/(1 + x^2) = dt
We'll re-write the integral in the new variable:
Int e^(arctan x) dx/(1 + x^2) =Int e^t*dt
Int e^t*dt = e^t + C
Int e^(arctan x) dx/(1 + x^2) = e^(arctan x) + C
The primitive function is: y = e^(arctan x) + C