What is the function y if dy/dx=(e^lnx)/x?
The function y has to be determined given that dy/dx = (e^lnx)/x
y = Int [ (e^ln x)/x dx]
let ln x = z
dz = dx/x
=> Int [ e^z dz]
=> e^z + C
substitute z = ln x
=> e^(ln x) + C
The function y = e^(ln x) + C
To determine the original function, we'll have to evaluate the indefinite integral of dy.
We'll replace ln x by t.
ln x = t
We'll differentiate both sides and we'll get:
dx/x = dt
We'll re-write the integral in the changed variable:
Int (e^ln x) dx/ x = Int e^t*dt
Int e^t*dt = e^t + C
The primitive function is: y = e^ln x + C