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October 19th, 2017, 08:44 PM   #1
woo
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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.
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October 19th, 2017, 11:03 PM   #2
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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.
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October 20th, 2017, 02:21 AM   #3
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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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