4 ( ) ( ) ( ) ( ) 4 5 5 II III A B (0 ) 4, 6, 7 II III A B (0 ) ( ),, 6, 8, 9 II III A B (0 ) ( [ ] ) 5, 0, II A B (90 ) log x x () (a) y x + x (b) y sin (x + ) () (a) (b) (c) (d) 0 e π 0 x x x + dx e 4x e x + dx x log x dx (cos x sin x 4 ) sin 5x dx a ( ) ( ) + sin x + sin x ( y log log a 4 0 x π ) a a
OABC OA OC 4, OB, AOB BOC COA 60 OA a OB b OC c () AB AC () ABC D OD ABC OD OA + p AB + q AC p q () OABC 4 f(x) + x ( ), () y f(x) P l 4 () y f(x) l P Q x () y f(x) l 5 5 5 4 k () () k () k (4) k 6 n n n + n + () n () k k 0 k n
7 () I O I X R XY P OP R XP YP O P II A E B I O C Y D () II ABC O I ABC R r AI ABC A D ABC AB E (A) (B) (C) (A) DB DI (B) AI DI Rr (C) OI R Rr 8 C : y (x t) + t C : y x + 4 t () C C t () () C C 9 ABC P PAB PBC PCA S AB S BC S CA A () P ABC S AB + S CA S BC BAC () S AB S BC S CA P ABC B P C
4 0 {a n } a, a n+ a n (n,,, ) () a a a 4 () a n y x y kx + (k ) P Q P x α Q x β (α < β) () α + β αβ k () P Q l m R R x k () () l m k
() (a) y x + x + x y ( + x ) ( + x ) 4x ( + x ) () (a) (b) y {sin(x + )} {sin(x + )} (b) (c) 0 e sin (x + ) cos(x + ) 6 sin (x + ) cos(x + ) x x x + dx (x x + ) x x + [ log(x x + ) e 4x e x + dx 0 0 e x (e x + ) e x dx e x + {e x (ex + ) e x + ] log 5 } [ ] dx ex log(e x + ) 0 (e ) log(e + ) + log x log e e ( ) x dx x log x dx 4 x log x dx [ ] e e 4 x log x 4 x (log x) dx e 4 e 4 x dx [ ] e 4 e 8 x 8 (e + ) (d) cos x sin x ( + cos x) sin x sin x + sin x cos x sin x + (sin 5x + sin x) cos x sin x 4 sin 5x sin x + 4 sin x π ( ) ( 0 sin x + ) 4 sin x dx [ 6 cos x 4 ] π cos x 0 4
6 t log ( + sin x) A log a 0 x π 0 t log + sin x a log 4 + sin a log ( + sin x) log a t A log + sin x a {log ( + sin x) log a } (t A ) ( ) ( ) + sin x + sin log log a 4 a (i) A + < 0 (t A)(t A ) 0 < a < {t (A + )t + A(A + )} { ( t A + )} 8 t 0 A(A + ) (log a)(log a + ) (ii) 0 A + a t log a 8 < a (iii) < A + t A(A ) (log a)(log a ) ( (log a)(log a + ) 0 < a < ) ( a 8 (log a)(log a ) ( < a) )
7 () a c 4 b AOB BOC COA 60 () a b a b cos 60 a c a c cos 60 4 6 4 4 8 AB AC ( b a) ( c a) b c a b a c + a 6 6 8 + 4 8 b c b c cos 60 4 6 OA AB a ( b a) a b a 6 4 0 OA AC a ( c a) a c a 8 4 8 AB b a b a b + a 6 + 4 AC c a c a c + a 4 8 + 4 6 () OD AB OD AB 0 ( OA + p AB + q AC) AB 0 OA AB + p AB + q AB AC 0 0 + p + 8q 0 OD AC OD AC 0 ( OA + p AB + q AC) 0 OA AC + p AB AC + q AC 0 8 + 8p + 6q 0 p q 6 () ABC AB AC ( AB AC) 6 8 6 () OD OA + AB + AC 6 OD OA + 4 9 AB + 6 AC + 4 OA AB + OA AC + AB AC 9 4 + 4 9 + 6 6 4 + + + 8 ( 0) ( 8) 9 8 OD OABC ABC OD 6 4
8 4 () y + x y x ( + x ) x y 8 l ( ), P l 4 y 4 8 () y + x y ( x ) y 8 x + 5 8 8 x + 5 8 y Q y 4 O P x + x 8 x + 5 8 x 5x + x + 0 (x ) ( x + ) 0 P Q x () x ( ) + x 8 x + 5 (x ) ( x + ) 8 8( + x ) S S + x dx ( ) 8 x + 5 dx 8 0 dx + x x x tan θ dx cos θ dθ θ π π 6 x θ π + x dx ( ) 8 + 5 8 S π π 6 + tan θ cos θ dθ [ dx 6 x + 5 ] 8 x π π 6 dθ π
9 5 () 5!!!! ( ) ( ) ( ) 0 4 () 4 ( ) k X Y Z x y z (x y z) k x y z x y z k X Y Z 0 0 5 0 5! ( ) ( ) ( ) 0 4 0 0 0!!! 8 () k (i) (ii) (i) x 0 y z 4 X Y Z 6 6 5! ( ) ( ) 4 0!4! 8 (ii) x y z X Y Z 5! ( ) ( ) 4 0!!! 8 0 8 + 0 8 40 8 (4) () () k 0 ( 0 8 + 40 ) 8 8 0 8 + 40 8 + 0 8 80 k 0 40 0 P (k) 8 8 8 8
0 6 () n n + n+c n+c n+c n+c (n + )n (n + )(n + ) n n + () k k + A k n k () A k P (k) (k 0,,,, n ) P (k) n k C n+c E(n) (n k) (n + )(n + ) n n (n k) E(n) kp (k) k (n + )(n + ) k0 k0 n (n k) k (n + )(n + ) k (n + )(n + ) n (nk k ) k { n (n + )(n + ) (n + )(n + ) n(n + )(n ) n(n ) (n + ) } n(n + ) n(n + )(n + ) 6 n(n ) E(n) > (n + ) > n(n 4) > 6 ( ) n 4 n(n 4) 0 n 5 n(n 4) 5 n 6 n(n 4) ( ) n 6
7 () OP S T XP YP SP TP (R OP)(R + OP) R OP OP R XP YP I X T II A P O E I O S Y B D C () (A) ABC A α B β CAD CBD CD CAD CBD DBI α + β DIB AIB DIB IAB + IBA α + β DBI DB DI (B) r IE AIE AI sin α r DB ABD sin α R DI (A) DB DI sin α R 4 4 AI DI Rr (C) I X Y P II A D I () OI R AI DI (B) OI R Rr
8 () C : y (x t) + t C : y x + 4 y (x t) + t x + 4 x tx + t + t 4 0 ( ) ( ) D D/4 ( t) (t + t 4) t t + 8 (t + 4)(t ) C C D > 0 (t + 4)(t ) > 0 4 < t < () ( ) α β α + β t, αβ t + t 4 C C (α, α + 4) (β, β + 4) P(x, y) x α + β t y ( α + 4) + ( β + 4) t + (t + t 4) + 8 (α + β) + αβ + 8 t + 4 t y x + () < t y < y x + ( < x < ) O x
9 () a BC b CA c AB ABC r S AB cr, S BC ar, S CA br S AB + S CA S BC ( ) ( ) ( ) cr + br ar c + b a BAC 90 () AP BC M BP CA N BM : MC S AB : S CA, CN : NA S BC : S AB P A S AB N S BC S AB S CA S AB S BC S CA M N B M C BC CA P AM BN P ABC BM MC CA AN NP PB NP PB S CA S AB + S BC (S AB + S BC ) PN + S CAPB 0 PA + S ABPC (S AB + S BC ) SBC + S CAPB 0 S AB + S BC S BCPA + SCAPB + SABPC 0 ABC P α PA + β PB + γ PC 0 S BC : S CA : S AB α : β : γ P( p) A( a) B( b) C( c) p α a + β b + γ c α + β + γ
4 0 () a a a 4 a a a 7 7 5 7 7 5 5 () () a n n ( ) n+ (i) n a n ( ) (ii) n k ( ) a k+ a k k k+ k+ ( k+ ) ( k ) k+ k+ n k + ( ) (i) (ii) n ( ) a n+ c a n c c, a n+ (a n ), a n+ a n a n a n a n+ a n+ a n a n an a n n a a a n n n+ n 4 n+
5 () y x y kx + y x kx + x kx 0 ( ) ( ) α β α + β k, αβ () y x y x P(α, α ) l y α α(x α) + α y αx α Q(β, β ) m y βx β x α + β y αβ ( ) k () R, R x k () () α < β β α (α + β) 4αβ k + 4 k y kx + x ( k S, k ( ) k RS + ( ) (k + 4) PQR RS (β α) 4 (k + 4) ) + () PQ y (α + β)x αβ PQ y x S S β α β [{(α + β)x αβ} x ]dx α (x α)(x β)dx y x l P α y kx + y O S R β Q m x 6 (β α) 6 (k + 4) PQR S 4 (k +4) 6 (k +4) (k + 4)