Integration of linear transform of function: Difference between revisions
(Created page with "==Statement== Suppose <math>F</math> is an antiderivative for <math>f</math>. Then: <math>\int f(mx + \varphi) \, dx = \frac{1}{m}F(mx + \varphi)</math> where <math>m</mat...") |
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Suppose <math>F</math> is an [[antiderivative]] for <math>f</math>. Then: | Suppose <math>F</math> is an [[antiderivative]] for <math>f</math>. Then: | ||
<math>\int f(mx + \varphi) \, dx = \frac{1}{m}F(mx + \varphi)</math> | <math>\int f(mx + \varphi) \, dx = \frac{1}{m}F(mx + \varphi) + C</math> | ||
where <math>m</math> is a nonzero real number and <math>\varphi</math> is a (possibly zero and possibly nonzero) real number). | where <math>m</math> is a nonzero real number and <math>\varphi</math> 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 <math>u = mx + \varphi</math>. | This is a special case of [[integration by u-substitution]] where we put in <math>u = mx + \varphi</math>. | ||
Latest revision as of 02:01, 19 December 2011
Statement
Suppose is an antiderivative for . Then:
where is a nonzero real number and 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 .