problem solving logarithmic equation and reaching an equivalence ok so i've had a problem trying to simplify the ln[sqrt(1+(u^2)/(a^2))+(u)/(a)] and this is supposed to be equal to : ln [ sqrt(a^2+u^2) + u ]
Aleah Booth
Answered question
2022-07-19
problem solving logarithmic equation and reaching an equivalence ok so i've had a problem trying to simplify the and this is supposed to be equal to : how is this posible ?? i've tried to solve this for more than 2 hours and couldn't get to this equivalence. any suggestions ?
Answer & Explanation
yermarvg
Beginner2022-07-20Added 19 answers
I expect that you got this as the result of an (indefinite) integration, and is a constant. Let be positive. We are taking the ln of
Taking the ln, we get
But is a constant, so can be absorbed into the constant of integration. In more detail, if
is the answer to an indefinite integral problem, where is an arbitrary constant, then
is a correct answer to the same problem. This sort of thing happens a lot, particularly with trigonometric functions. As a simple example, if is "the" answer to an indefinite integration problem, then so is
Aleah Booth
Beginner2022-07-21Added 5 answers
is injective so, if . So, the following should hold true: