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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.

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  • $\begingroup$ There is always a constant when you are dealing with indefinite integrals. You must have only seen improperly done examples or definite integrals. $\endgroup$ – Rocket Man Nov 12 '15 at 17:47
  • $\begingroup$ Do we get a constant when we integrate by parts? $\endgroup$ – SHASHANK PATHAK Nov 12 '15 at 17:49
  • $\begingroup$ Yes, you do. If $F(x)$ is the antiderivative of $f(x)$ where $F(x)$ was found using integration by parts, then $F(x)+c$ is also an antiderivative of $f(x)$. To check, just differentiate $F(x)+c$ and see what you get. $\endgroup$ – Rocket Man Nov 12 '15 at 17:53
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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.

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

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

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