Editing 2117: Differentiation and Integration
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'''{{w|Integration by parts}}''' | '''{{w|Integration by parts}}''' | ||
β | The "product rule" run backwards. Since <math>(uv)' = uv' + u'v</math>, it follows that by integrating both sides you get <math> uv = \int u dv + \int v du</math>, which is more commonly written as <math>\int u dv = uv - \int v du</math>. By finding appropriate values for functions <math>u, v</math> such that your problem is in the form <math>\int u dv</math>, your problem ''may'' be simplified. The catch is, there exists no algorithm for determining what functions they might possibly be, so this approach quickly devolves into a guessing game - this has been the topic of an earlier comic, | + | The "product rule" run backwards. Since <math>(uv)' = uv' + u'v</math>, it follows that by integrating both sides you get <math> uv = \int u dv + \int v du</math>, which is more commonly written as <math>\int u dv = uv - \int v du</math>. By finding appropriate values for functions <math>u, v</math> such that your problem is in the form <math>\int u dv</math>, your problem ''may'' be simplified. The catch is, there exists no algorithm for determining what functions they might possibly be, so this approach quickly devolves into a guessing game - this has been the topic of an earlier comic, {{1201: Integration by Parts}}. |
'''{{w|Integration by substitution|Substitution}}''' | '''{{w|Integration by substitution|Substitution}}''' |