October 19th, 2017, 08:44 PM  #1 
Newbie Joined: Jan 2014 Posts: 19 Thanks: 0  Notation for indefinite integrals
The notation $\int{f(x)}\,dx$ is commonly used to denote the set of ALL antiderivatives of the function $f$. My question: Is it wrong to use the notation $\int{f(x)}\,dx$ to denote a single antiderivative of the function $f$ without the constant of integration $c$? I saw many instances where the notation $\int{f(x)}\,dx$ is use to denote a single antiderivative with the constant of integration omitted. For example to evaluate $\int {xe^x}\,dx$, some textbooks wrote $v=\int\,dv=\int{e^x}\,dt=e^x$, without the constant of integration. For example, to solve the differential equation $y'+\frac{2}{t}y=e^t$ by using an integrating factor, some textbooks wrote $\mu(t)=\exp(\int{2/t}\,dt)=t^2$, without a constant of integration. 
October 19th, 2017, 11:03 PM  #2 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,838 Thanks: 653 Math Focus: Yet to find out. 
Try introducing a constant at that step. You will end up with a sum of multiple of arbitrary constants which is still arbitrary, and we simply 'recycle' them.

October 20th, 2017, 02:21 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 20,966 Thanks: 2216 
If the integrating factor is found as the exponential function of an antiderivative, omitting the arbitrary constant doesn't matter, as a valid integrating factor results, regardless of the value of the constant.


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