29

9

.,,, 3 () C k k C k C + C + C + + C 8 + C 9 + C k C + C + C + C 3 + C 4 + C 5 + + 45 + + + 5 + + 9 + 4 + 4 + 5 4 C k k k ( + ) 4 C k k ( k)

3 n( ) n n n ( ) n ( ) n 3 ( ) 3 3 3 n 4 ( ) 4 4 4 ( ) n n + 3 4 + n + x + x x + x x 3 + x 4 x x + x dx [log + x ]x log + x log log log + x x x ( x + x x 3 + )dx x x + x3 3 x4 4 + log + x x x + x3 3 x4 4 +

4 x log + + 3 3 4 4 + log + 3 4 + ( ) n n n log

() 5 4, 4,, 5 : 4 6 8.8 { 5.4 4..8.8.8 5 7

6 (3) 6, 3 7 6 x 6 3 7 7 3 : 7 3 3 4 5 6 7 8 9 7-3 4 5 6-3 - 3 3 4 5-3 7 5 46 48 : 3 69 : 9 97 : 6 5 8 : 4 38 39 : 6 : 84 88 : 5 7 9 : 3 6 6 6 84 84 84 5 + 3 8 + 8 6

(4) ABC, BC, CA, AB D, E, F 3 AD, BE, CF 7 : OABC BD DC CE EA AF F B b c c a a b

8 (5) sin x + cos x > 5 4 ( x π ) a sin θ + b cos θ a + b sin(θ + α) ( ) + 5 5 4 5 ( 5 sin x + 5 cos x cos θ 5, sin θ 5 ) > 5 4 θ 5 5 (sin x cos θ + cos x sin θ) > 4 5 5 {sin(x + θ)} > 4 sin(x + θ) >

9 sin(x + θ) > π 6 x + θ 5 6 π π θ 6 θ x 5 6 π θ x x π π 6 θ x π cos θ θ arc cos 5 5 π 6 arc cos 5 x π...9.8.7.6.5.4.3.. π θ 6 y sin x + cos x y 5 4 π 6 arcsin( 5 ).5995 π :

{log ( x + )} + log ( x + ) 6 > {log ( x + )} + log ( x + ) 6 > log {log ( x + )} + log ( x + ) 6 > log log {log ( x + )} log ( x + ) 6 > t log ( x + ) t t 6 > (t 4t ) > (t + )(t 6) > t, 6 t <, 6 < t t log ( x + ) x + > > x + 4 > x + 3 > x () { log ( x + ) < () log ( x + ) > 6 (3)

() log ( x + ) < log ( x + ) < log x + < x + < 4 4 < x + 7 4 < x + 49 6 < x + 49 6 < x 33 6 < x (3) log ( x + ) > 6 log ( x + ) > log 6 x + > 6 x + > 64 64 > x + 6 > x + x + 6 33 6 < x < 3

( (6) z 3(cos θ + i sin θ) < θ < π ), z i z, z, P, P, P z O, P, P, P, z z c r(cos φ + i sin φ) re iφ z z c z z c z z c z c z a + ib z zz (a + bi)(a bi) a + b z r zz r z a + ib, z a + ib z + z (a + a ) + i(b + b ) z + z (a + a ) i(b + b ) z + z z z c z c (z z c )(z z c ) z c z c z z z z c z c z + z c z c z c z c z z z z c + z z c () z z c z c (z z c )(z z c ) z c z c z z z z c z c z + z c z c z c z c z z z z c + z z c () z z c z c (z z c )(z z c ) z c z c z z z z c z c z + z c z c z c z c z z z z c + z z c (3)

3 z 3(cos θ + i sin θ) 3e iθ a + bi ( ) a a + b a + b + i b a + b a + b (cos θ + i sin θ) a b cos θ, sin θ a + b a + b z i z ( + i)z 4 + + i ( ) 4 z { } ( i) z { } ( i) ( i ) z z ( cos π 4 i sin π ) z 4 e i π 4 z e i π 4 3e iθ 3 (e i π 4 e iθ ) 3 e i(θ π 4 ) z z 3 e iθ

4 z 3e iθ, z 3 e i(θ π 4 ), z 3 e iθ () z z 3e iθ 3e iθ 9 z z c 3e iθ re iφ 3re i(θ φ) z z c 3e iθ re iφ 3re i(θ φ) 9 3re i(θ φ) + 3re i(θ φ) 3r(e i(θ φ) + e i(θ φ) ) 6r cos(θ φ) (4) () z z 3 e i(θ π 4 ) 3 e i(θ π 4 ) 9 z z c 3 e i(θ π 4 ) re iφ 3 re i(θ φ π 4 ) z z c 3 e i(θ π 4 ) re iφ 3 re i(θ φ π 4 ) 9 3 re i(θ φ π 4 ) + 3 re i(θ φ π 4 ) cos θ + i sin θ i sin θ 3 r {cos(θ φ π 4 ) + cos(θ φ π } 4 ) 6 { r cos (θ φ) π } 4 cos(α β) 6 r { cos(θ φ) cos π 4 + sin(θ φ) sin π 4 6 [ ] r {cos(θ φ) + sin(θ φ)} 6 r{cos(θ φ) + sin(θ φ)} } 3r{cos(θ φ) + sin(θ φ)} (5)

5 (3) z z 3 e iθ ( 3 eiθ ) 9 z z c 3 e iθ re iφ 3 re i(θ+φ) z z c 3 eiθ re iφ 3 rei(θ+φ) 9 3 re i(θ+φ) + ( ) 3 rei(θ+φ) 3 r ( e i(θ+φ) + e i(θ+φ)) r cos(θ + φ) (6) 3 (4) 9 6r cos(θ φ) cos(θ φ) 3 r (7) sin (θ φ) + cos (θ φ) sin (θ φ) cos (θ φ) sin(θ φ) ± cos (θ φ) ± 9 (8) 4r (7) (8) (5) 9 3r{cos(θ φ) + sin(θ φ)} 9 3r 3 r ± 9 4r 3 r 3 r ± 9 4r

6 sin(θ φ) sin(θ φ) θ φ, π 9 4r < θ < π θ φ θ φ θ φ cos(θ φ) 3 r cos 3 r 3 r r 3 θ φ r (6) 9 r cos(θ + φ) 3 9 3 3 cos θ cos θ 9 cos θ 9 cos θ 8 9 cos θ 4 9 cos θ ± 3 < θ < π cos θ 3

7 sin θ + cos θ sin θ cos θ sin θ 4 9 sin θ 5 9 sin θ 5 3 z ( cos φ sin ) φ 5 z 3 3 + i 3 + 5i

8. () x ix i x, a + bi a, b x ix i x ix + ( i) i ( ) x ix + ( i) (x i) ( ) i ( cos π + i sin π ) e i π (x i) e i π x i ± e i π 4 ± ( cos π 4 + i sin π ) 4 ± ( + i ) ±( + i) x i ±( + i) x i ± ( + i) i ± ± i ± + ( ± )i a + bi x ± + ( ± )i { + i

9 (),,, x i ± (i) 4( i) i ± i,, a + bi,,, n (cos θ + i sin θ) n cos nθ + i sin nθ cos(nθ) + i sin(nθ) (e iθ ) n e inθ i (cos θ + i sin θ) cos θ + i sin θ i i cos θ + isin θ { cos θ sin θ

θ θ π 4 i (cos θ + i sin θ) ( cos π 4 + i sin π ) 4 ( + ) i x i ± i i ± i i ± ( + ) i i ± i ± ( + i) { + i ( + ) i

3. OABC O, A, B, C, OA a, OB b, OC c a 4, b 9, a b 6, b c 4, c a 8, 3: OABC () c a, b c α a + β b c a (α a + β b ) a α a + β a b 8 6α + 6β b c b (α a + β b ) α a b + β b 4 6α + 9β

α β { 8 6α + 6β 4 6α + 9β 8 6α + 6β )4 6α + 9β 6 3β β 8 6α + 6β β 8 6α + 6 6α 3 8 6α 4 α 3 α β c α a + β b 3 c a + b

3 () AB OC D, BC OA E, OB L, AC M, DE N,, 3 L, M, N, LM : LN 4: OABC OD AB OD a + s( b a ) ( s) a + s b OD OC OD t c 3t a + t b

4 OD s 3t a s t b s 3t t 3t 3t t t s t t s t t s 4 OD 3 a + 4 b

5 OE BC OE c + s( b c ) ( s) c + s b ( ( s) 3 ) a + b + s b ( 3 ) 3s a + b + a s b + s b 3 (s ) a + ( s) b OE OA OE t a OE 3 (s ) t a s b s s 3 (s ) t s 3 ( ) t t 3 OE 3 a

6 L OB b OL M AC a + c OM a + ( 3 a + b ) a + b 4 a + b N DE ON ( OD + OE) ( 3 a + 4 b + 3 ) a ( 3 ) a + 4 b 3 4 a + b

7 LM : LN LM OM OL ( ) b a + b 4 b a + 4 MN ON ( OM 3 ) a + b 4 a + b LM ( ) a + b 4 MN LM L, M, N LN LN ON OL ( 3 ) b a + b 4 3 b a + 4 3 LM L, M, N LM : LN : 3

8 4. xyz, C : z x a + y, a >, D : x + y 4 () C x E P 3,, E Q P Q P Q xy R, R 5: C 6: D C x z y E Q (, t, t ) ( < t < ) P Q (x, y, z) (x, y, z) (3,, ) + s((, t, t ) (3,, )) (3( s), st, + s(t ))

9 R xy z + s(t ) s(t ) s t s t x 3( s) ( 3 + ) t 3(t ) t + 3 t 3t t y st t t x 3t t, y t t x(t ) 3t t 3t x y(t ) t t t y 3t x t y 3t y x t 3t y t x 3t y x t x 3y

3 x 3( s), s y t x 3( s) x 3 s x 3 s x 3 y t ( ) x t 3 y t y x 3 t 3y x 3 x 3y 3y x 3 x(x 3) 3y 3y x(x 3) 9y (3, )

3 x(x 3) 9y x(x 3) 9y x 3x 9y ( x 3x + 9 ) 9y 9 4 4 ( x ) 3 9y 9 4 ( x 3 ) 9y 9 4 ( ) x 3 9 4 ( x 3 9 ) y 4 4y 4 ( x 3 9 4 ( ) x 3 9 4 ) ( x 3 ) 4 y 4 4y 4 4 ( 3 y ) ( )

3 () C D z V a V 7: C D ( ) 4 x x dx 4 x a + y dy 4 4 ( ) 4 x x 4 dx a + y dy [ x 4 dx a y + ] 4 x 3 y3 { } 4 dx a x (4 x ) + 3 (4 x ) 3 [{ } { }] 4 a x (4 x ) dx + 3 (4 x ) 3 dx [ { } 4 x (4 x ) a dx + { }] (4 x ) 3 dx 3

33 x (4 x ) dx (4 x ) 3 dx x sin θ dx cos θdθ sin θ sin θ sin θ sin θ π θ x (4 x ) dx π 4 π π π π π π ( sin θ) {4 ( sin θ) } cos θdθ 4 sin θ {4 (4 sin θ)} cos θdθ 4 sin θ {4( sin θ)} cos θdθ cos θ 4 sin θ (4 cos θ) cos θdθ 4 sin θ cos θ cos θdθ 4 sin θ 4 cos θdθ 4 sin θ cos θdθ π 4 ( sin θ cos θ) dθ

34 sin θ cos θ sin θ x (4 x ) dx 4 π 4 (sin θ) dθ [ θ 8 sin 4θ ] π {( 4 π ) 8 sin 4 π ( π 4 4 ) 8 sin π } 4 π 4 π

35 (4 x ) 3 π dx {4 ( sin θ) } 3 cos θdθ 4 π π π π π π π π (4 4 sin θ) 3 cos θdθ {4( sin θ)} 3 cos θdθ (4 cos θ) 3 cos θdθ 3 cos 3 θ cos θdθ 3 cos 3 θ cos θdθ 4 cos 4 θdθ ( cos θ) dθ ( cos θ) dθ cos θ cos θ + π 4 (cos θ + ) dθ π 4 (cos θ + cos θ + )dθ [ 4 θ + 8 sin 4θ + ] π sin θ + θ [ 3 4 θ + 8 sin 4θ + ] π sin θ {( 3 4 π + 8 sin 4 π + ) sin π ( 3π 4 4 + 8 sin π + ) sin π ( ) 3π 4 4 + + 3π }

36 [ { } V 4 x (4 x ) a dx + { }] (4 x ) 3 dx 3 ( 4 a π + ) 3 3π ( ) π 4 a + π ( ) 4π a +

37 5. () {a n }, ε >, N, N < n, N < m a n a m < ε n, a n, {a n} k k n > m ( a n a m + + ) + + m n + + + m m+ n n k km+ m n m > n ( a m a n + + ) + + n m + + + n n+ m m k kn+ n m a n a m m n a n a m m n ( + + ) m ( + + ) n ε < N < ε N N < n N < m ( ) a n a m < Max, n m < < N {a n }

38 () n n+ n > n N n n dx x N+ n > N+ x dx dx [log x]n+ x log(n + ) log log(n + ) N N n lim N n n n lim N N n lim N N n n n > lim log(n + ) N + n > + n

39 n n dx x n n n dx x n n n 8: