The trigonometric functions and are nice functions since their derivatives and antiderivatives are easy to obtain. So for the integration by parts, these functions have the same behavior whether we differentiate them or take their antiderivatives. Therefore the focus should be on the other function involved in the integration. In this case, we must differentiate because its derivative gives the constant 1.
It is a common mistake to forget the constant .
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Mohamed A. Khamsi