MATH2402FDF.pdf - ′(ln | = =(ln ∙ ∙ ′ ′ log = ∙(ln Basic Integration Rules 1 � = 1 ≠ −1 1 of the top and bottom and evaluate the limit

MATH2402FDF.pdf - ′(ln | = =(ln ∙ ∙ ′ ′ log =...

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𝑑𝑑 𝑑𝑑𝑑𝑑 (ln | 𝑢𝑢 |) = 𝑢𝑢′ 𝑢𝑢 𝑑𝑑 𝑑𝑑𝑑𝑑 ( 𝑎𝑎 𝑢𝑢 ) = (ln 𝑎𝑎 ) ∙ 𝑎𝑎 𝑢𝑢 ∙ 𝑢𝑢′ 𝑑𝑑 𝑑𝑑𝑑𝑑 log 𝑎𝑎 𝑢𝑢 = 𝑢𝑢 𝑢𝑢 ∙ (ln 𝑎𝑎 ) Basic Integration Rules : � 𝑑𝑑 𝑛𝑛 𝑑𝑑𝑑𝑑 = 1 𝑛𝑛 + 1 𝑑𝑑 𝑛𝑛+1 + 𝐶𝐶 , 𝑛𝑛 ≠ − 1 sin 𝑢𝑢 𝑑𝑑𝑢𝑢 = cos 𝑢𝑢 + 𝐶𝐶 cos 𝑢𝑢 𝑑𝑑𝑢𝑢 = sin 𝑢𝑢 + 𝐶𝐶 tan 𝑢𝑢 𝑑𝑑𝑢𝑢 = ln | cos 𝑢𝑢 | + 𝐶𝐶 cot 𝑢𝑢 𝑑𝑑𝑢𝑢 = ln | sin 𝑢𝑢 | + 𝐶𝐶 sec 𝑢𝑢 𝑑𝑑𝑢𝑢 = ln | sec 𝑢𝑢 + tan 𝑢𝑢 | + 𝐶𝐶 csc 𝑢𝑢 𝑑𝑑𝑢𝑢 = ln | csc 𝑢𝑢 + cot 𝑢𝑢 | + 𝐶𝐶 � 𝑒𝑒 𝑢𝑢 𝑑𝑑𝑑𝑑 = 𝑒𝑒 𝑢𝑢 + 𝐶𝐶 𝑢𝑢′ √𝑎𝑎 2 − 𝑢𝑢 2 = sin −1 𝑢𝑢 𝑎𝑎 + 𝐶𝐶 𝑢𝑢′ 𝑢𝑢√𝑢𝑢 2 − 𝑎𝑎 2 = 1 𝑎𝑎 sec −1 | 𝑢𝑢 | 𝑎𝑎 + 𝐶𝐶 𝑢𝑢′ 𝑎𝑎 2 + 𝑢𝑢 2 = 1 𝑎𝑎 tan −1 𝑢𝑢 𝑎𝑎 + 𝐶𝐶 Integration by Substitution : ∫ 𝑓𝑓�𝑔𝑔 ( 𝑑𝑑 ) �𝑑𝑑𝑑𝑑 = 𝑓𝑓 ( 𝑢𝑢 ) 𝑑𝑑𝑢𝑢 𝑔𝑔 ( 𝑏𝑏 ) 𝑔𝑔 ( 𝑎𝑎 ) 𝑏𝑏 𝑎𝑎 , where 𝑢𝑢 = 𝑔𝑔 ( 𝑑𝑑 ) and 𝑑𝑑𝑢𝑢 = 𝑔𝑔′ ( 𝑑𝑑 ) 𝑑𝑑𝑑𝑑 L’Hopital’s Rule : When taking a limit, if you get an indeterminate form i.e. ± ± , 0 0 , then take the derivative of the top and bottom and evaluate the limit again. Integration by Parts : ∫ 𝑢𝑢𝑑𝑑𝑢𝑢 = 𝑢𝑢𝑢𝑢 − ∫ 𝑢𝑢𝑑𝑑𝑢𝑢 , where 𝑢𝑢 = ∫ 𝑑𝑑𝑢𝑢 Trig Substitution :
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