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Math 180 Derivatives and Integrals Basic Differentiation Rules d cu cu ' dx d u vu'uv ' 4. dx v v2 d x 1 7. dx d u e e u u' 10. dx d sin u (cos u ) u ' 13. dx 16. d csc u ( csc u cot u ) u ' dx d 19. arcsin u u' 2 dx 1 u 1. d u v u ' v' dx d c 0 5. dx d 8. u u u ' dx u 2. d u a a u (ln a)u ' dx d cos u ( sin u ) u ' 14. dx d sec u (sec u tan u ) u ' 17. dx 11. 20. d uv uv' vu' dx d n u nu n 1u ' 6. dx d ln u 1 u ' u ' 9. dx u u d 12. log a u 1 u' u' dx u (ln a) u (ln a) d tan u (sec 2 u ) u ' 15. dx d cot u ( csc 2 u ) u ' 18. dx 3. d arccos u u' 2 dx 1 u 21. d arctan u u ' 2 dx 1 u Basic Integration Rules 1. 3. k f u du k f u du du u C u n1 C ; n 1 n 1 7. e u du e u C 5. n u du sin u du cos u C 11. tan u du ln cos u C 13. csc u du ln csc u cot u C 15. sec u du tan u C 17. sec u tan u du sec u C 9. 2 19. a 2 du 1 u arctan C 2 a a u 2. 4. [ f u g u ] du f u du g u du k du k u C du ln u C u 1 au C C 8. a u du a u ln a ln a 10. cos u du sin u C 6. 1 u du 12. cot u du ln sin u C 14. sec u du ln sec u tan u C 16. csc 2 u du cot u C 18. csc u cot u du csc u C 20. du a2 u2 arcsin u C a Integration by Parts: u dv uv v du Arc Length: b S 1 f ' ( x) dx 2 a Math 190 Formula Sheet Powers of the Trig Functions: 1) Integrals of the form: sin m x cos n x dx m or n odd Strategy: If m is odd, save a sine factor and convert to cosine If n is odd, save a cosine factor and convert to sine 2) Integrals of the form: sin m x cos n x dx m and n even and non-negative Strategy: use ½ angle identities: 1 cos( 2 x) 1 cos( 2 x) cos 2 x ; sin 2 x 2 2 3) Integrals of the form: sec m x tan n x dx ; if m is even Strategy: Save a sec 2 x and convert to tangent 4) Integrals of the form: sec m x tan n x dx ; if n is odd Strategy: Save a secx tanx and convert to secant 5) tan n x dx n any positive integer Strategy: convert a tan2 x to sec2x-1 and distribute; repeat if necessary 6) sec m x dx ; m is odd Strategy: Integrate by parts *** If all else fails convert everything to sines and cosines Trigonometric Substitution: Form Trig Sub Identity x a tan 1 tan 2 sec 2 a2 x2 a2 x2 x2 a2 x a sin 1 sin 2 cos 2 x a sec sec 2 1 tan 2 Trigonometric Identities: sin 2 x cos 2 x 1 sec 2 x 1 tan 2 x csc 2 x 1 cot 2 x 1 tan x 1 sec x cos x 1 csc x sin x cot x sin 2x 2 sin x cos x cos 2 x cos 2 x sin 2 x Numerical Integration Approximations: ba f x0 2 f x1 2 f x2 ... 2 f xn1 f xn TRAP (n) 2n ba f x0 4 f x1 2 f x2 4 f x3 ... 2 f xn2 4 f xn1 f xn SIMP (n) 3n