( ) 2.1. C. (1) x 4 dx = 1 5 x5 + C 1 (2) x dx = x 2 dx = x 1 + C = 1 2 x + C xdx (3) = x dx = 3 x C (4) (x + 1) 3 dx = (x 3 + 3x 2 + 3x +

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Download "( ) 2.1. C. (1) x 4 dx = 1 5 x5 + C 1 (2) x dx = x 2 dx = x 1 + C = 1 2 x + C xdx (3) = x dx = 3 x C (4) (x + 1) 3 dx = (x 3 + 3x 2 + 3x +"

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1 (.. C. ( d C ( d d + C + C d ( d + C ( ( + d ( d C (5 9 + d + d tan + C cos (sin (6 sin d d log sin + C sin + (7 + + d ( d log( C ( (8 d 7 6 d C ( (9 ( d d ( d ( ( ( ( log + C ( d d + + C ( ( + d d + d d d d C d C log + C

2 ( (5 (6 (7 (8 (9 ( + d ( + d log + C ( d d 8 + C ( + d ( + d d + + C ( sin cos d cos sin + C cos cos d ( cos cos d tan sin + C (e d e log + C ( ( d 7 + d C.. C. ( sin d ( cos d cos + cos d ( ( ( (5 e d ( + cos d cos + sin + C ( e d e + e d ( + e + C + + ( ( + sin sin d sin d sin + cos + C ( log d log d log d log + C log( + d ( + log( + d ( + log( + d ( + log( + + C

3 (6 ( e d ( e e d ( e + C (7 (8 log d log ( sin d ( ( cos d log 9 + C cos d ( cos sin + C (9 ( I a d a log a log a e cos d. a d a log a a (log a + C I e cos d e cos + e sin d ( e cos + e sin e cos d ( e cos d e e (sin + cos I (sin + cos + C. sin d sin sin d cos sin + cos sin cos d cos sin + sin ( sin d cos sin cos sin d sin d sin cos cos + C ( ( log( + + d log( + d + log( + d + log( + + C

4 ( I sin 5 d. I sin 5 d sin sin d cos sin + cos sin cos d cos sin + ( sin sin d cos sin + sin d I sin 5 d 5 sin cos + sin d 5 5 sin cos + ( 5 sin cos cos + C 5 sin cos 5 sin cos 8 5 cos + C ( (5 (6 (7 e + d e + e + d ( e + + C ( log( + d log( + d log( + ( + d log( + ( d log( + ( + C e log(e + d (e + log(e + d (e + log(e + d (e e + e + d (e + log(e + d e d (e + log(e + d e + C sin( d cos( + cos( d cos( + sin(

5 (8 5 log d 6 6 log 6 5 d 6 6 log C 5 (9 I I cos d. cos cos d sin cos sin cos ( sin d sin cos + cos d cos d sin cos + sin I cos d sin cos + sin + C. (. sin d cos + cos d ( cos + sin cos + sin 6 sin d ( cos + cos + sin + 6 cos 6 sin + C cos d.. ( log t d. log d log t + C log log + C t ( + + d d. + + d d. + t ( + + t + tan t + C tan ( + + C

6 6 ( + 5 d d. t ( + d. t d t + t t + + t + log(t + + tan t + C log( tan + C ( t d, d. cos d cos t sin t + C sin + C (5 sin d cos d. t d, d. sin d cos t t sin t + C sin + C + cos (6 cos d d. t d, d. cos d + cos t t + sin t + C + sin + C (7 t d, d. e d e t et + C e + C (8 sin t cos d. cos e sin d e t e t + C e sin + C (9 t d, d. d 9 t t sin + C sin + C

7 ( log( + t log( + + d + d, + d. t t + C { log( + } + C ( t d, d. d t t + C ( + C ( t d, d. ( t ( d t t 6t + 9 t ( 6t + 9t 7 ( t + 6 log t + 9 t e ( + e + e d d log( + e + C + e + C + 6 log C ( + t d, d. ( + d t 5 t5 + C 5 ( C (5 t d, d. t d ( + C (6 + d ( + + d log + + C (7 5 + t 5d, d. 5 (5 + d 5 t t + C (5 + + C (8 + t d, d. e + d e t et + C e+ + C

8 8 (9 π t d, d. ( π sin d sin t cos t + C ( π cos + C ( t t +. d, d. d t + t ( t + t t t + C ( ( + C ( {( + 5} + C 5 ( ( + + C ( + t t. d. + d (t t ( t t 5 t 5 t + C 5 t (t 5 + C 5 ( + ( + C ( t d, d. d (t + t ( + 9 t t + log t + C 9 9 ( + log + C 9

9 9 ( t d. (t + d t ( t + t + t 5 t 5 + t + t + C 5 t ( t + t C 5 ( C ( + t d. + d t t + C ( + + C (5 t d, d. d t 9 t + C 9 ( + C (6 sin t cos d. sin cos d t t + C sin + C (7 t d, d. ( d t 5 t 5 + C ( 5 + C (8 t d, d. cos d cos t tan t + C tan + C

10 (9 + t d, d. ( + d t t (t t ( 5 t5 t + C 8 t (t 5 + C 8 (8 ( + + C ( + t d, d. d t + t (t t ( t t + C t (t 6 + C ( C ( t 6d, d. 6 d 6 t t 6 t + C ( + C ( t 9 d, d. 9 e d e t 9 9 et + C 9 e + C

11 .. ( cos t sin d, sin d. sin ( cos d t t + C cos + C ( t d. + d t + tan t + C tan + C ( e t e t +, t. t + e t d t t + t t + ( t + t tan t + C e d, d e e tan e + C ( sin t d cos t. d sin t cos t cos t t + + cos t sin t + C sin + sin(sin cos(sin + C sin + + C (5 sin t cos d. sin cos d t t + C sin + C

12 (6 cos t sin d, sin d. cos t sin d t + C cos + C.5. ( ( + ( ( + 5 A + + B + ( + ( ( + 5 C + 5. A( ( B( + ( C( + ( A, B, 5 C 8. ( ( ( ( + ( ( + 5 ( + + ( + 8( + 5 ( ( A + B + ( ( C. A( ( + B( + C( A, B, C. ( ( + ( ( + A + B + C + ( ( +. A( + + (B + C( A, A C C. A + B B. ( ( ( + A + B + C + ( +. + A( + + (B + C (A + B + C + A A, C, B. + ( + +

13 (5 ( + ( + A + + B + C +. ( + ( + A( + + (B + C( + A, A + C C. A + B B. ( + ( + ( + ( + (6 + ( + A + B + C + ( +. + A( + + (B + C (A + B + C + A A, C, B. + ( + + (7 (8 ( + ( + A + + B + C + ( + ( +. A( + + (B + C( + A, A + C C. A + B B. ( + ( ( + ( + A + + B + C + ( + ( A( + + (B + C( + A, A + C C. A + B B. 5 + ( + (

14 (9 ( ( ( ( + A + B + + ( + C ( +. A( + + B( + + C A, C. A + B B. ( + + ( + ( ( A + B + ( ( C (. A( + B( ( + C( A, C. A + B B. ( ( + ( 7 ( + ( A + + B + ( + ( C (. 7 A( + B( + ( + C( + C, A. A + B B. 7 ( + ( + + ( + ( + ( ( + A +. ( B ( + C A( ( + + B( + + C( ( + + D( D ( + B, D, A + B + C + D, A + C. A + C. A, C. + ( ( + ( + ( + + ( +

15 5 ( ( ( + A + B + + C + D +. A( + ( + + B( ( + + (C + D( A, B, A B D, D. A + B + C, C. ( ( + + ( + ( ( ( + A + B + + C + D +. A( + ( + + B( ( + + (C + D( A, B, A B D, D. A + B + C, C. ( ( + ( + (5 ( ( + + A + A( (B + C( B + C + +. (6 A, A C, C. A + B, B. ( + ( ( + ( A + B + + ( + ( C + D A( + ( B( (C + D( +

16 6 A, B. A + B + C, C. 8A + B + C + D, D ( + ( (7 ( + A + B + ( + + C + D ( +. (A + B( + + C + D A + B + (A + C + B + D (8 A, B, C, D. ( + + ( + 5 ( + ( + A + + B ( + + C + D +. ( + ( + 5 A( + ( + + B( + + (C + D( + B, A+B +D, A + D 5. A + C + D. A + C. A, C, D. 5 ( + ( ( (9 ( + ( + A + B + + C + D +. ( + ( + (A + B( + + (C + D( + (A + C + (B + D + (A + C + (B + D A + C, A + C, B + D, B + D A C, B, D. ( + ( + ( + ( +

17 7 ( + ( + ( + A A( + + (B + C( + B + C +. A, A + C (, C. A + B B. + ( + ( + + ( + ( + A A( + + (B + C( + B + C +. A, A + C (, C. A + B B. + ( + + ( + + ( ( + A + B + ( ( + C +. + A( ( + + B( + + C( A, B, C. + ( ( + + ( + ( A + B ( + + ( + + C + D ( (A + B( + + C + D A + B + (A + C + B + D A B C, D ( ( +

18 8 ( + ( A + B + C + D ( A( B( (C + D (A + C + (A + B + D + (5A + B + 5B B 5, A 5, D 9 5, C 5. + ( ( ( A ( ( + (6 B ( + + C ( A( + + B( + + C A, B, C. A + (A + B + (A + B + C ( ( + + ( ( ( + ( ( + A + B + + C D A( + ( ( + + B( ( ( + + C( ( + ( + + D( ( + ( A, B, C, D ( ( ( ( + A + B + + C + D +. + A( + + B( + + (C + D (A + C + (B + D + A + B A, B, C, D

19 (8 ( ( + ( ( A + + B + + C. + A( ( + B( + ( + C( + ( A, B, C ( + ( A A( (B + C( + B + C A, 5A + C, C. A + B B ( ( A ( A( + + B( + + C A + (A + B + (A + B + C A, B, C ( + + ( + ( + ( 5 A + + B 5. ( + ( 5 A( 5 + B( + B ( + + C ( +. A 6, 5 B 6. ( + ( 5 6( 5 6( +

20 , { } ( + ( 5 d 6( 5 d 6( + 6 log 5 6 log + + C ( 6 7 ( + ( 7 A A( 7 + B( + B 7. A 8, 7 B 8., 6 7 8( 7 8( + { } 6 7 d 8( 7 d 8( + 8 log 7 8 log + + C ( ( + ( + A A( + + B( + B +. A, B , ( d + + d log + log + + C ( 9 ( ( + A + 9 B +. A( + + B( A 6, B ( 6( +

21 , { } 9 d 6( d 6( + 6 log 6 log + + C (5 ( + ( + A + + B +. ( + ( + A( + + B( + A, B., ( + ( + ( + ( + { } ( + ( + d ( + d ( + log + log + + C ( ( +, d ( + d + + C (7 (8 + 5 ( 5, + 5 d 6 + d ( + d t d. 6 + d t + ( 5 d 5 + C tan t + C tan ( + C

22 (9 ( + + d ( + + d + t d. + + d t + tan t + C tan ( + + C d ( + + d + t d d t + tan t + C tan ( + + C ( + A + B +. + A( + + B( A, B , ( + d + d + log + log + + C ( ( ( A A( + B( B. B, A 5., ( ( { } + + d 5 ( d ( 5 log log + C

23 ( + ( + + d d log( C ( + ( + ( A + + B + 5. A( + + B( A, B. + ( + ( +, { } + d ( + d ( + log + log + + C (5 6 ( ( + A + 6 B +. A( + + B( A,, 6 + ( 6 d + B. d log + log + + C (6 8 + d ( + + d log( + + C

24 (7 (8 (9 ( d ( + d + t d. + t d t ( t + t log t t + C log C 6 + d ( d log( C ( 7( + A 7 + B A( + + B( 7 7 A 9 8, B 7 8., ( ( + { } d 9 8( d 8( log log + + C 8 + d ( + d t d. t + + d t + ( t t + + t + ( (t + + t + t + log(t + + tan t + C log( + + tan ( + C

25 ( ( ( ( + 5 d ( + 5 d + 5 log( C d + 6 ( + + d + t d. + 6 t d t + ( t t + + t + ( (t + + t + t + log(t + + tan t + C log( tan ( + + C 6 d ( 6 6 d log 6 + C + 5 d ( 5 d 5 t d. t d t ( t + t 5 (5 (6 (7 + 5 d + + d { + log t t + C log C ( + 5 { + + } d + 6 ( + ( + A + + B ( + + C + + ( + ( + log C } d + log + + C D ( +. A( + ( + + B( + + C( + ( + + D( + B 5, D 7, A + C, A + C.

26 6 (8 (9 ( ( A, C, ( + ( ( ( + ( + ( + d + d + 5 { + 5 ( + } + 7 d ( + log log C ( 6 + d log C ( ( A A( + B( B A, B 7., ( d d + + ( + ( A + + B + ( + ( d + log 7 log + C ( + 7 d log + + C C (. + A( + B( + ( + C( + A 9, C 5, A + B B 9. + ( + ( 9( + + 9( + 5 (

27 7, + ( + ( d { } 9( + + 9( + 5 d ( 9 log log 5 ( + C ( ( + ( ( + 5 A + + B + ( + ( ( + 5 C + 5. A( ( B( + ( C( + ( A, B, 5 C 8. ( + ( ( + 5 ( + + ( + 8( + 5, ( + ( ( + 5 d { } ( + + ( + d 8( + 5 ( ( log + + log + log C ( 5 d d log + C + ( ( + + A + B + C + +. ( ( A( (B + C( A 5, C 5, A + B B 5. + ( ( + + 5( + 5( + +, { + ( ( + + d 5( + 5( log 5 } d + ( + + d

28 8 + d + t. d, ( + + t + ( + + d t + t t + t + log(t + tan t + C log( + + tan ( + + C ( 5 log 5 log( tan ( + + C (5 + + ( ( + ( + A + B + + C + D + + A( + ( + + B( ( + + (C + D( ( +. A, B, A B D D,, A + B + C C. +, + { d ( + ( + + ( + } ( + ( + + d ( + log + log + + tan + C (6 + ( ( + A + B + C + ( ( + + D + E ( +. + A( + + (B + C( ( + + (D + E( (A + B (B C + (A + B C + D + ( B + C D + E + (A C E A + B B C A + B C + D B + C D + E A C E

29 9 A, B C, D, E. { + ( ( + d ( + ( + + } d ( + { log ( + + ( ( + } d log 8 log( + tan ( + d ( + d ( + d ( + d + t d. t t + C ( + + C,.6 ( + d ( + + tan + C ( log 8 log( + tan ( + d ( + d log 8 log( + tan + ( + ( + tan + C log 8 log( + tan ( + tan + C.7. ( cos t sin d, sin d. (cos sin d ( t t t + C cos + cos + C ( sin t cos d. cos sin + d t + tan t + C tan (sin + C

30 ( sin t cos d. (sin + sin + cos d (t + t + t + t + t + C sin + sin + sin + C ( sin t cos d. cos sin d t t + C sin + C (5 cos t sin d, sin d. {sin(cos } sin d sin t cos t + C cos(cos + C (6 cos t sin d, sin d. (cos 5 cos + 7 sin d (t 5t + 7 ( t 5 t + 7t + C cos + 5 cos 7 cos + C (7 cos t sin d, sin d. sin cos + d log t + + C log( + cos + C t + (8 cos t sin d, sin d. sin cos d sin cos d t t + C cos + C (9 tan t cos d ( + tan d d + t

31 t (tan + tan + t + + d t + ( t + t t + t + t + t log(t + tan t + C tan + tan log(tan + tan (tan + C tan + tan + log cos tan (tan + C ( tan t cos d ( + tan d d + t ( tan t tan d t ( + t ( t t + t tan t + C tan tan (tan + C cos d ( + tan d d + t tan d (t (t + d { ( t ( t + (t + log t log } log t + tan t + C tan tan + tan (tan + C

32 ( (sin + cos d ( + sin cos d ( + sin d cos + C. ( sin sin d sin d sin cos d. cos t sin d. sin d t ( t t + log t t + + C log cos cos + + C ( tan t t. cos + t, d. + t + cos d + t +t + t t + C tan + C (5 tan t t t. sin, cos + t + t, d. + t + sin ( + cos sin d + t +t ( + t ( t + + t t +t +t + t t + t + log t + C tan + tan tan + log + C (6 cos sin (cos + sin cos + sin d d log cos + sin + C cos + sin

33 (7 tan t t. sin + t, d. + t + sin d + t + t +t t + t + (t + + t + tan + C (8 cos sin + d ( tan tan + + C cos sin + d log(sin + sin + sin + d tan. + t sin + d t + +t (t + t. sin t + t t + + C tan + + C cos sin + d log(sin + + tan + + C sin + d + t, d

34 .8. ( 6 + d d. + t t 6, d 6t 5. + d 6t 5 t + t 6t t + ( 6 6 t t + t + ( t t + t log t + t t + 6t 6 log t + + C + C ( + t t, d t. t d + t t + C + + C ( t. t +, d t. t + d t t (t + t 5 t5 + t + C 5 ( 5 + ( + C ( t. t + t t. t t + ( t, d t t t t + t t t

35 5 d ( t + t t t t (t + t + u t du. ( u du u + C t t t + + C ( C + + C (5 t. t + t t. t t t t + t t d t t t t 8t + 6 t ( t 8 t + 6 t 8 t log t t + C t + t, d ( t t t 8 ( + + log + + ( C 8 ( + log ( + C log C

36 6 (6 ( + t. t, d t. ( + d ( t t t 8 t t + t t t t 5 + t 6t ( t t + t 6 ( 6 t 5 t 5 + t + C 8 ( + 9 ( ( + + C 9 ( + { 5( + 9( } + C 9 ( + ( C (7 + t. t, d t. + d (t t t (t t 5 t5 t + C 5 ( + 5 ( + + C 5 ( + {( + 5} 5 ( + ( + C

37 (8 d d ( t. t +, d d t t + d t. (t + t t + 7 t + C (9 + 9 t. t 9 t t + 9., t + 9 d + 9 t t t 9 t t 9 t t +9 t t + 9 t t 9 ( t t + + C t 9 ( t, d + 9 t t + 9 t log t log t + + C log log C ( 5 + t. t5, d 5 t. 5 d + t 5 t 5 t 5 8 t + C 5 8 ( C

38 8 ( + + d ( + d { } ( + + C { } ( + + C ( + t. t t t +., t + d + t t t t t t t + t t + t t ( t, d + t t ( t t + log t log t + + C t + t log + + log C ( t. t, d t. d ( t t ( t t ( t t + log t log t + + C log log + + C ( + + t. t t + t (t +, { } d + (t + t +., (t + (t t t t t + t + t + t +

39 9 + + d (t + t + t t +t+ (t + t+ t+ t ( t t + log t log t + + C log log C (5 + d d + + ( + t. t t + t +, d, ( t + t + t (t +. + d + ( + t + t + d t (t + t (t + (t + t (t + (t + A t + +. (t + (t + B (t + + Ct + D t + t A(t + (t + + B(t + + (Ct + D(t + t B, t A + B + D A + D. t A + C. t A + B + C + D A + C + D. A, C,

40 D. ( ( t + (t + tan t + t + + C + tan + + C tan + + C tan C (6 n + t. t n, d nt n. n + d (t n t nt n n (t n t n n n + tn+ n n + tn+ + C n n+ ( + n n n+ ( + n + C n + n + n n+ ( + n {(n + ( + (n + } + C (n + (n + n n+ ( + n {(n + n} + C (n + (n + (7 + t. t, d t. + + d + t t ( t + t log t + + C + log( C

41 (8 + 7 t. t 7, d t d t (t 7t t t t t 7t + 6 ( { { t 7t + 6 (t (t (t + (t (t (t + } } { + t 5(t 5(t + t + log t } log t + + C 5 log t log log log C (9 t. t + t t., t t t ( t + t t d t t t t t + t ( t t + t 8 t log t 8t + C t + t, d ( t t t 8 ( + log + 8( + + C 8 ( + log + ( log + + C 8 + C

42 ( t d. d t t + C ( + C.9. ( ( ( ( (5 (6 (7 e + π π 6 ( + + d [ ( + + ] d ( 6 ( d [log ]e + log e log ( d + + d [ ] + + log ( + 8 ( + log log ( d ( + d + d ( + d [ ] ( [ (+ d ( + ( d + d + [ ] + log + + log (sin t cos t [ cos t sin t] π π 6 ( ( ]

43 (8 + t [ tan t ] tan π (9 d [ sin ] sin π 6 ( ( π cos θ dθ π ( + d + cos θ dθ ( d + [ θ + ] π sin θ π [log log + ] (log log ( log log log ( d ( ( + d ( + d [log log + ] (log log ( log log + log.. ( t + t ( d d [ log( + t ] log( + log t + t d d { log( + } log + ( e t F (t + C. d d ( + e t d d d d [F (t]+ {F ( + F (} e(+

44 ( F (t + C. t + ( d d t + d [F (t] d d d { F ( F ( } 6 + ( ( tf(t f(t tf(t g( g(t G(t + C., d d ( g(t [G(t] G( G( g(t d d (G( G( G ( g( g(t f(t g(t tf(t ( d ( tf(t d ( f(t tf(t d d f(t + f( f( f(t... (. log(cos t ( lim lim log(cos ( lim sin ( d d d d ( lim t + ( cos e t. e t d d d d [F (t] 8 + e t F (t + C (F ( F ( e e

45 5. lim ( e t lim e e. ( ( ( ( (5 (6 (7 (8 e π π e e log d [ log ] e cos d [ sin ] π e π ] a d [ a a log a log a d a [ ] log a a (log a a log a d e [] e e e + sin d [cos ] π a (log a + (log a a log a a + (log a sin d [ cos ] π π + [ ] e e log d log d [ ] e e e e e e d [e ] tan d [ tan ] cos d [sin ] π e d e [e ] e e + π [ log( + (log d [ (log ] e e e e log d ( e [ log ] e + d ] π log (log d e d e (e [] e e (e e + e

46 6 (9 ( π e sin d π sin sin d [ cos sin ] π π π + sin d [ ] e log d log [ e π sin ( sin d 6 π e ] e cos sin cos d π sin d sin d [ cos ] π d π sin d ( ( + d e 6 e e + 6 [ ] ( + 5 ( + d [ ] ( sin d [ sin ] d d t.. d, t d t sin d [ sin ] π t [ t] d ( π + (

47 7. ( t d, t. [ ] d t ( t t ( t d, t π π. π sin d ( t d, e d π sin t t. [ cos t]π e t [et ] e e ( d ( ( + d. ( ( + A +. ( ( + A( + + B( B + A, B. d ( ( + d ( + [ log d log + (log log 5 (log log log log 5 ] d

48 8 ( d + ( + ( + d + ( + ( + A + +. ( + ( + B + + A( + + B( + A, B d + ( + ( + d ( + + d [ log + log + ] ( log log ( log log log log (6 (7 (5 + + d (5 + ( + d [ ] (5 + ( + 5 ( + d (7 [ ] 5 ( + 5 (7 8 ( 5 8 ( + d [ 5 ( ] { 5 ( } 5 ( + 5 d [ ( ( ] (8 t d, t. d t ( t [t ]

49 9 (9 e d t. d, t. (e e d e d e t ( e d e d e t e d e t ( π π sin d t. d, π π π t π π. π sin d π π sin t, π π [ cos t]π π {cos π cos( π} { ( } cos d s. d ds, π π cos d π π cos sds [sin s] π π π π s π π. ( π π + d ( ( (sin cos d ( + d + [ tan ] log( + ( π log ( π log π

50 5 ( 9 d 9 d + 9 d. 9 d t. d, 9 d 9 d ( t 9t 9 d + t 9t + t 9t 9 d 9 d t. ( t d, t. e d e t [et ] (e ( + t d, t. + d t [ ] t ( (5 + + d [ ( + d ( + ( ] (8 (6 + t d, t d 5 [ 5 t 5 t 5 ] ( 5 5 5

51 5 (7 t d, t. d t t [ ] t (8 t t d t, π. t π π sin d π t sin t [ t cos t] π π + cos t [sin t] π.. ( ( π π π π sin 6 d π π sin 6 d π π π sin 6 d + π π sin 6 d + π π sin 6 d + π sin 6 d. π sin 6 d t π d, π π π sin 6 d π π cos d ( sin 6 t + π π π π sin 6 d sin 6 d π π cos d + π cos 6 t sin 6 d π π π sin 6 d t π sin 6 d. sin 6 d 5 6 π 5 8 π. π cos d. π cos d. t d, π π cos d π cos d π π t π cos ( t π. π π cos d + cos d + π cos t π π cos d

52 5 π π π π cos d cos d t π d, π π π ( cos t + π π π sin t t π cos t., π π π π cos d s π ds d, cos d π π cos (s + πd cos sds π π s π. π π cos d π π π π cos d + π cos d π cos d cos d + cos d π π cos d ( π π sin 5 d π sin 5 d +, t d, π sin 5 d. π t π. π sin 5 d π π sin 5 d π π π sin 5 d + π sin 5 ( t( + sin 5 t + π sin 5 d π sin 5 d sin 5 d (.5. π sin π cos sin + cos d cos sin d cos + sin

53 5 π sin cos sin + cos d π π π sin cos sin + cos d + π sin cos (sin + cos sin + cos sin d 8 [ cos ] π d cos sin cos + sin d.5. ( t π d, t π π. π π sin d π π π π sin d π π π (π t sin (π t (π t sin t sin t π π sin t π cos t [ t sin t ] π t sin t π ( tan t d cos t ( + tan t, t π. π log( + d + π π π log( + tan t ( cos t + sin t log cos t log cos(t π cos t log + π ( log cos t π π log cos t

54 5 π ( log cos t π t π s ds, t π s π. π log cos ( t π π log + (log [] π + π 8 log π π π log cos( s( ds π ( log cos t π π log cos sds π log cos t log cos sds log cos t.6. ( lim n n ( lim n n ( lim n lim n n lim n ( lim n ( e n + e n n + + e n e d [e ] e (sin πn + sin n π + + sin nn π sin πd [ ] π cos π π ( n n ( n + n n n n n n + n n n n n n d [ + ] + ( n + log n + ( n n lim n n n + log n + { ( n lim n n + log + n n log( + + d + n + log n + n + n n + log n log n + + n + n log n + n n + + n n n + n log n + n n ( + n n n ( log + n } n

55 55 log( + t + d, t log..7. log t ( < π < sin < π. π d < π π [ ] log t (log π d < sin d [] π π, d d [ ] π π π π < sin d < π ( < < < <, d < d < d [] d [sin ] π 6 < d < π 6 d ( (, ( d d ( d

56 56 d sin t d cos t, t π, d π π cos t π d [ t + + cos t π (. sin t ] π π e e d e d [ e e d [] ( e e d d ] ( e e d.8. ( d + d + d [tan ] π π d [] + d d

57 57,. + d ( ( ( + ( + d ( + {( + } + [log( + ] 6 [log( + ] + log + 9 π ( k < < k + k+ k (k + < k {( + } [ tan d (k + < <. k (k + d < k+ k k+ k d < k d < k,,..., n n n k+ (k + < d < n k k+ k n+ d n k ( k < < k + k+ k k k k+ k n k k k d [ d ] n+ n + (k + < n + < n k k < <. k + k k+ d < k + k+ k k+ d < k < d < k + k k k,,..., n n k k + < n k k+ k d < n k k k d ( ]

58 58 n k k+ k n+ d n k ( k < < k k k k + < k d [ ] n+ ( n + k + < ( n + < k + d < k n k k + < + < k k k k + d < + d < k k k k k k k. k d k,,..., n n n k k + < n + d < k n k k k + d n k n k + < [ + d log(n + < n ] n log( + k k log(n +.9. ( ( ( M [ e d lim e d lim ] e M ( lim M M e M M + d lim M M. d lim ( M M [ d lim ] M + + M lim M d lim ( M lim M ( + M ] M + [ ( M +

59 59 ( (5 (6 (7 (8 M d lim + M d lim ( M N + M lim M N +. d lim ( ( ( N M M lim M lim M lim M [ d lim + tan ] M M lim M tan M π [ d lim ( M N + + ( M N M ( d ( [ log ] M ( log M d lim M N N + M lim M N + lim M N + lim M N +. d lim M N + lim M N + lim M N + lim M N + d M log log M ] M N ( ( d ( N d [ ] log M N ( log M M log N N M N M N d ( + d [ sin ( + ] M N { sin (M + sin (N + } π

60 6 (9 d lim M lim M M d + [ ] M d d + lim + lim lim (M + M N + N + [ + lim N + N ] N d ( N 6 ( + + d [ d sin ] π ( ( e log d,. lim e M + {[ M lim M + lim M + lim M + log ] e e } d log [ M ] M e log ( M e e M log M M e lim M + M log M lim M log M M + lim M + lim M + lim M + log M M M M ( M ( + M d lim M + d sin t d cos t, M t sin M.

61 6 N sin M + sin M d lim M lim M lim M lim N π lim N π lim N π sin M sin M N + sin t sin t ( + sin t ( + ( [ t sin t ( N cos t cos t cos t ] N sin N π ( π tan d π lim M π π tan d + tan d π M π tan d + lim tan d N π + N lim M π [ log cos ]M + lim N π +[ log cos ]π N lim log cos M + M π ( lim log cos N + N π +.. ( log d lim log d + M M { [ lim M lim + lim N + [ M lim M log d log( log d + lim N + d N ] M log( } M d { [ ] log } d N N ] M [ 9 log( + lim N + ( M 9 log( M M 9 + ( 9 M 9 log M + M + lim N + log 9 ] N

62 6 (5 (6, lim + lim + d ( + log. log lim + lim M + M lim + d lim M log 5 d M ( lim + [ ] ( lim lim M M + M M + ( + d ( + A + B + C +. ( + A( + + (B + C A, A+B+C B + C, A + B C B C. B, C. d lim ( + M lim M lim M lim M M M ( + d ( + d [ log log( + ( log ] M M log + M log (7 tan d lim M M { [ lim M lim M lim M tan d ] M M } ( + d tan + [ tan + log log( + ( M tan M + log π + log ] M M + + M tan log

63 . ( + d log log( + + C (6 6 (8 tan d lim M N + lim M N + lim M N + lim M N + lim M N + M N tan d { [ tan ] M M N N { [ tan ] M M + N N [ tan + log( + {( M tan. + } + d ] M M + log(m + ( N tan N + log(n + N ( } d } +.. s d lim s M M [ ] M d lim s M s s. (i s >. s <, d s s (ii s <. s >,, s d lim M M d. s, d lim s M + M d s. d lim [log M ]M [ lim M + ( lim M s M s s d s d. s ] s s M lim M log M ( lim M + s s M s

64 6 (i s <. s >, d s s (ii s >. s <,, s.. d lim M + M d d. s lim [log M + ] M d s d. s lim ( log M + M + ( y + y. (, y ( y + y ( ( +,,., y + y +. {( + ( + }d ( + d [ + ( y y ( ( + ],,,.,.

65 65 ( d + [ ] ( + d ( y y + [ + ] 7 6 (,,.,. ( d ( + y. [ ] ( d ( + d ] [ + 6

66 66 (5 y log y. + e e ( log d + [] e log d { + [] e [ log ] e e + [] e [ log ] e e (6 y +. } d [ ( + d (7 y + y + ] (, ±,. 6 5

67 67 (8 y ( + d [ ] 5 + y. ( + [ d tan ] π (9 y sin. π sin d [ cos ] π ( y log e. e log d [ log ] e e d e [] e

68 68 ( y e y ( e,. y + y e, (. lim e, lim e., y e. e d [ e ] ( e d [ e e ] e ( y + y,. ( + y + y,. lim ± +, y + y ( + [ d log( + ] log

69 69 ( y sin y sin π sin sin sin cos sin sin ( cos sin, cos, π, π, y sin y sin. π (sin sin d + [ cos + cos ] π π π + (sin sin d [ cos + cos ] π π ( y ( (. 5 ( ( d ( + d [ + ] ( ( d ( + d [ + ] 7

70 7 (5 y (. ( d ( + d [ + ] (6 y + y, y. y +. y + + y y + y +, + + ( (,, lim ( + ±, lim ( + ±. ± ± y +.

71 7 ( + [ d + log ] log (7 y sin ( + cos y (cos + ( cos. y sin ( + cos. π π y + + y y sin ( + cos. π sin ( + cos d π (sin + sin d [ cos ] π cos

72 7 (8 + y. sin t d cos t, π d π π π d. t π π cos t cos t + cos t [t + ] π sin t π. π.. ( y log y. y log (c, log c y log c c ( c y c + log c c e, y. e y log.

73 7 e e ( + e log d [ ] e { e } e + e [ log ] e d ( e e + [ log ] e e e ( y log y log. y log c y log c c log c c ( c y log c c log c c. c e, y e. y log. e y + y e. e + e ( e log d e [ ] e e + e e log d log d

74 7 log t e d, t. ( t [ t ] 8 ( y y. A y, B y., (,.. ( + + d + [ ] ( + [ 9 + ] d ( t y t t, >, (t t [ t ] t 8 ( t y t t, <, [ ] (t t t t t

75 ( θ π y cos θ, cos θ,, π ( cos θ dθ π π ( cos θ + cos θdθ ( cos θ + [ θ sin θ + sin θ cos θ dθ ] π π ( θ π. θ π y cos θ, sin θ cos θ..6 π cos θ sin θ cos θdθ π π cos θ( cos θdθ (cos θ cos 6 θdθ ( π 5 6 π 8 π 75.. ( r sin θ (θ, r-. r >. θ r (, y-.

76 76 π ( sin θ dθ π π sin θdθ ( cos θdθ [ θ sin θ ] π π ( r sin θ (θ, r-. θ r (, y-. π sin θdθ π ( cos θdθ [θ sin θ ] π π ( r cos θ (θ, r-.

77 θ r (, y-. 77 π ( cos θ dθ π π π ( cos θ + cos θdθ ( + cos θ cos θ + dθ ( cos θ + cos θ dθ [ θ sin θ + sin θ ] π π ( r cos θ (θ, r-. r. θ r (, y-. π [ cos θdθ sin θ ] π

78 78.5. ( y, t + t + t t + + d. + t t t t d ( + t + t t t +., + t t t t t + t + t + + d t + t t + (t + t + t [ t + log t ] + t [ ( + + log( + ] 8 ( + [( + + log( + ( ] 8 { + log( + } ( y sin cos tan, π π + tan d π cos d cos cos d π cos sin d

79 79 sin t cos d, t ( t t + π t. [log t log t + ] log( + ( cos θ sin θ, y sin θ cos θ, π ( cos θ sin θ + ( sin θ cos θ dθ π π sin θ cos θdθ π sin θ cos θdθ 6 sin θdθ [ cos θ] π 6 ( e t cos πt πe t sin πt, y e t sin πt + πe t cos πt, (et cos πt πe t sin πt + (e t sin πt + πe t cos πt + π.6. ( r θ, π e t + π [e t ] + π (e π θ + (θ dθ θ + θ dθ + θ t θdθ, θ t π + π. π θ + θ dθ +π [ t t ] +π 8 {( + π }

80 8 ( r sin θ, π π ( + cos θ + ( sin θ dθ + cos θdθ π π cos θ dθ cos θ π dθ cos θ dθ [ sin θ ] π π [ sin θ ] π π 8 ( r sin θ cos θ, π sin 6 θ ( + sin θ cos θ π dθ sin θ dθ π π sin θ dθ cos θ dθ [θ sin ] π θ π ( r e θ, π e θ + e θ dθ π e θ dθ (e π (5 r cos θ ( sin θ θ cos sin θ cos θ 8 + θ θ π cos6 sin dθ cos θ 6 dθ π π cos θ π dθ cos θ, ( sin θ dθ sin θ t cos θ dθ, θ π t. ( t [ t ] t 8

81 8 (6 r e θ, e θ + e θ dθ lim M + M e θ dθ [ lim ] e θ M M + ( lim e M M ,..,..,... (...., y f(. a b b b ( + {f (} d + f ( d a.. (... dy (t, y y(t d y (t., a b α t β (t..5 b β ( y + f (t β ( d + (t (t a α (t + y (t α.. (... r f(θ, (, y { (θ r cos θ f(θ cos θ y(θ r sin θ f(θ sin θ a { (θ f (θ cos θ f(θ sin θ y(θ f (θ sin θ + f(θ cos θ. α θ β β α β α {f (θ cos θ f(θ sin θ} + {f (θ sin θ + f(θ cos θ} dθ f(θ + f (θ dθ..

82 8 (... { r cos θ y f( (, y y r sin θ. r sin θ f(r cos θ. r θ θ r sin θ + r cos θ f (r cos θ {r cos θ r sin θ} r {sin θ f (r cos θ cos θ} r {cos θ + f (r cos θ sin θ} r r cos θ + f (r cos θ sin θ sin θ f (r cos θ cos θ r + (r r + r {cos θ + f (r cos θ sin θ} {sin θ f (r cos θ cos θ} r + f (r cos θ {sin θ f (r cos θ cos θ} α θ β. β β + f r + (r dθ r (r cos θ {sin θ f (r cos θ cos θ} dθ α r cos θ a b α d (r cos θ r sin θdθ { r cos θ + f } (r cos θ sin θ cos θ r sin θ dθ sin θ f (r cos θ cos θ r sin θ f (r cos θ cos θ dθ, α θ β a b.5 b + f r (r cos θ {sin θ f (r cos θ cos θ} dθ a + f (d.. sin θ f (r cos θ cos θ r d.,.,...8. ( π [ ] 6 d π π

83 8, y, π + ( d π + 9 d + 9 t 6 d, t. π + 9 d π π 8 [ t t 6 ] π 7 ( ( π. π π cos d π π ( + cos d π [ + ] π sin π, y sin, π π cos + sin d sin t cos d, π t. π π cos + sin d π + t + t u t + t u t + t u tu + t, t u u ( + du u + du., u u + t u t u + u u u t u +,

84 8 π π π t u + u + du u u (u + u + u du [ π u + log u ] + u {( π + + log( + } ( + { π ( + + log( + ( } { } π + log( +

9 5 ( α+ ) = (α + ) α (log ) = α d = α C d = log + C C 5. () d = 4 d = C = C = 3 + C 3 () d = d = C = C = 3 + C 3 =

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