Diff and Integ Rules Final
Transcription
Differentiation Formula (chain rule form) Integration Formula for f (u) d ku ku' k = constant dx d kf (u) kf '(u) dx d f (u) g(u) f '(u) g'(u) dx du u C d u v u v ' v u' dx d u v u' u v ' dx v v2 u dv u v v du d n u nu n 1u' dx d sin u u' cos u dx d cos u u' sin u dx d tan u u' sec 2 u dx d sec u u' sec u tan u dx d cot u u' csc 2 u dx d csc u u' csc u cot u dx (First) Fundamental Theorem of Calculus: b f ( x) dx F (b) F (a) a Second Fundamental Theorem of Calculus: u = g(x) kf (u) du k f (u) du C f (u) g (u) du f (u) du g (u) du u n du u n 1 C n 1 (integration by parts) n 1 cos u du sin u C sin u du cos u C sec 2 u du tan u C sec u tan u du sec u C csc 2 u du cot u C csc u cot u du csc u C tan u du ln sec u C ln cos u C cot u du ln csc u C ln sin u C sec u du ln sec u tan u C ln sec u tan u C csc u du ln csc u cot u C ln csc u cot u C u f (t ) dt u ' f (u ) v' f (v) v where u g (x) and v h(x) 2 arcsin u du u arcsin u 1 u C arccos u du u arccos u 1 u C arctan u du u arctan u ln u 1 C arc cot du u arccot u ln u 1 C arcsec du u arcsec u ln u u 1 C 2 2 2 2 arccsc u du u arccsc u ln u u2 1 C d u' arcsin u dx 1 u2 d u' arctan u dx 1 u2 d u' arcsec u dx u u2 1 d arccos u dx d arccot u dx d arccsc u dx u 2 2 dx du 2 2 u u u 2 a2 dx du u u 2 a2 au C or ln a u u e du e C d u' log a u dx u ln a ln u du u ln u u C u C a a u a u u du 1 u a 2 u 2 dx a 2 u 2 a arctan a C d u a u'a u ln a dx d u e u'e u dx d u' ln u dx u arcsin u a du u 1 ua u dx u du ln u C u u 1 arcsec C a a ln a dx a u C
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