Integration of linear transform of function

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Statement

Suppose F is an antiderivative for f. Then:

\int f(mx + \varphi) \, dx = \frac{1}{m}F(mx + \varphi) + C

where m is a nonzero real number and \varphi is a (possibly zero and possibly nonzero) real number). The "+ C" is the usual arbitrary constant addition.

This is a special case of integration by u-substitution where we put in u = mx + \varphi.