Why is the integral of (5)/(1+x^2) not tan^-1(5x)?
In calculus, both in the case of finding the derivative and when we are finding the integral, with differentiation being the inverse of integration, the constant multiples of the terms with the variables is excluded and only the final answer is multiplied by the constant. The integral of c*f(x) = c*Int[ f(x) dx]
Here, we have the integral of f(x) = 5/(1 + x^2). As 5 is a constant it not included while the integral is being found.
Int[f(x) dx] = Int[5/(1 + x^2 dx]
=> 5*Int[5/(1 + x^2 dx]
=> 5* arc tan x + C
The correct right integral of 5/(1 + x^2) is 5*arc tan x + C.