What does the absence of constant in integration by parts signfy?

Question

While integrating by parts we do not get a constant whereas in other cases we do. Does this mean that integrating a function by parts does not give us a family of curves but a definite curve? Also if this is true then why is it special in case of integration by parts? Also many functions can be integrated both by parts and simple integration e.g f(x)=x. Why is there an ambiguity of constant? Please clarify.

Answer 1

The integration by parts formula states that $$\int u(x)v^{\prime}(x) dx = u(x)v(x) - \int v(x)u^{\prime}(x)dx.$$ Notice that, even though you don't see a constant yet, you still have an integration to perform. When you do that, you will get a constant.

Answer 2

It's typically suppressed because when you integrate by parts, you still have an antiderivative left in the expression. For example, you typically see

$$\int xe^x\;dx =\int x\;d(e^x) =xe^x-\int e^x\;dx=xe^x-e^x + C $$