# 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 $u = mx + \varphi$.