x3 + 1 (sin x)/x d dx (f(g(x))) = f ( g(x)) g (x). d dx (sin(x3 )) = cos(x 3 ) (3x 2 ). 3x 2 cos(x 3 )dx = sin(x 3 ) + C. d e (t2 +1) = e (t2 +1)

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Download "x3 + 1 (sin x)/x d dx (f(g(x))) = f ( g(x)) g (x). d dx (sin(x3 )) = cos(x 3 ) (3x 2 ). 3x 2 cos(x 3 )dx = sin(x 3 ) + C. d e (t2 +1) = e (t2 +1)"

Λευίς Κρεστενίτης
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1 x sin x cosx e x lnx x3 + (sin x)/x e x {}}{ (f(g(x))) = f ( g(x)) g (x). }{{}}{{} f(g(x)) 3x cos(x 3 ). 3x cos(x 3 ) x 3 3x sin(x 3 ) (sin(x3 )) = cos(x 3 ) (3x ). 3x cos(x 3 ) = sin(x 3 ) + C. e ( +). + e ( +) ( ) ( ) e ( +) = e ( +).

2 ( ) +) = +) = e e( e( ( +). e ( +) = e( +) + C. e( +) x 3 x + 5. x + 5 g(x) = x + 5 x 3/ /(3/) g (x) g(x), x (g(x)) 3/ 3/ = (x + 5) 3/. 3/ (x + 5) 3/ 3/ ( (x + 5) 3/ ) 3/ = 3 (x + 5) / x 3 = x 3 (x + 5) /, 3/ (x + 5) 3/ = 3/ 6 (x + 5) 3/. x 3 x + 5 = 6 (x + 5) 3/ + C. ( ) 6 (x + 5) 3/ = 6 3 (x + 5) / x 3 = x 3 (x + 5) /. = (x) = 3x cos(x 3 )

3 (x) 3x 3x cos(x 3 ) = cos(x 3 ) 3x }{{}}{{} = x 3. x 3 cos = sin + C = sin(x 3 ) + C. cos x = (/) f(g(x))g (x) g(x) f(x) F f F = f (F(g(x))) = f(g(x))g (x) f(g(x))g (x) = F(g(x)) + C. = g(x) / = g (x) f() = F() + C. F = f f() = F() + C. f() = f(). = (x) e ( +) + = +. = () =. = e ( +) = e ( {}}{ + ) }{{} = e = e = e + C = e( +) + C.

4 e = e + C C C C C x 3 x + 5 x + 5 x 3 x 3 = x + 5. = (x) = x 3, = x3. x 3 x + 5 = = / = 3/ 3/ + C = 6 (x + 5) 3/ + C. = x + 5 x x + 5 x = x 3 # # % e cos θ sin θ θ = cosθ sin θ sin θ = (θ)θ = sin θ θ, = sin θ θ. e cos θ sin θ θ = e ( ) = ( ) e = e + C = e cos θ + C.

5 e + e + e e = + e = e e + e = + e e = = ln + C = ln + e + C = ln( + e ) + C. ' ( ) * +, ( + e ) ) -. / ) ) 5, 6 7 e, = e. (/( + )) u = + an θ θ anθ = (sin θ)/(cos θ) = cosθ = sin θ θ sin θ anθ θ = cos θ θ = = ln + C = ln cosθ + C. xe x f(x) = xe x x = x = x = x xe x = e = e + C = + C. ex xe x = = ex (e e ) = (e ). xe x = e + C, x x x = x = = x = = x = = = x = x= x= xe x = = = e = e = ( e e ) = (e ).

6 π/ an 3 θ cos θ θ anθ cos θ θ (anθ) = cos (cos θ) = sin θ, θ θ /cos θ sin θ = anθ = (/ cos θ)θ θ = = an = θ = π/ = an(π/) = π/ an 3 θ cos θ θ = π/ (anθ) 3 cos θ θ = 3 = =. 3 5 x = 5 x = x = = x = 3 = 3 ( ) 5 x = = ln = (ln ln ) = ln = ln.69. = = = x = + x = x ( ) = x ( ) = + x = ( ) = / ( ) ( = ( 3/ / ) = 5 5/ ) 3 3/ + C ( = 5 ( + x) 5/ 3 ( + ) x) 3/ + C. = + x

7 (x + 7) 3 3 x x (x + 7) = 3 x = ( /) = x + 7 = 3 x x = 3 x = 3/ / x + 7 (x + 7) 3 ( 3 3 x = ) ( ) = ( 7 ) /3 = (7 ) /3 = (7 /3 /3 ) ) (7 /3 + C = = 3 /3 7/3 7/3 ( 5 (3 x)/3 3 (3 x)7/3 7 ) + C. 3 3 x b (/) a b a π/ x + x sin x cos x sin(x + ) sin(x 3 + ) xcos(x + ) x cos(x 3 + ) xsin(x + ) x sin(x 3 + ) cos( ) sin(3 ) (r + ) 3 r ( 3 3) x(x + 3) y ( + y) y sin(x) xe x y(y + 5) 8 y x ( + x 3 ) x(x ) 7/ ( 7) 73 e e x e 3x 5e. y y + 5 (x + 3) x x e x3 +

8 sin θ(cos θ + 5) 7 θ sin 6 θ cos θ θ sin 6 (5θ)cos(5θ) θ (ln z) z z y y + y e y y y e x + e x + 3 ( + ) cosh x (sinh z)e cosh z z xcosh x cos 3 sin3 sin 3 α cos α α an(x) e + e + cos x x + e x x + e x x + x + x + 9 e x e x e x + e x xcos(x ) sin(x ) sinh 3 cosh( + ) cosh xsinh x π π/ 8 3 cos(x + π) e cos θ sin θ θ e 3 x 3 x cos x x (x 3 + 5x) 3 / e 3 x x + y y + y cos(πx) xe x + x ( + x ) + y y ( + 7) sin z(z + ) /3 z p() = π 3 + f(x) = sin 3x f(x) = xcos(x ) r() = cos( 3 ) f(x) = sin( 5x) f(x) = e sin x cos x f(x) = x f(x) = x + 3cos (x) + + x x + + x (z + ) z z 3x x + π/3 3 sin (3x) = ln(s + ) s = π sin (y) y ln( + ) e π (ln ) 3 = (π x) cos x = b a = b a z 3 e z z π = x g() = e x b a e x = e b g() e a xcos(π x) < a < b a < b g() g()

9 π π cos θ sin θ θ e x cos x + e x + sin x ln x x x e e sin x arcsin x cos x x (sin x) 3 cos x x + x xsin(x ) + x x (x 3 + ) 3 x x + x + x x 3 e x x sin x x ( + x ) sin x + cos x f(x) = xe x x = x = x + f(x) = /(x + ) x = x = f(x) = sinh(x/) x = x = f(θ) = (e θ+ ) 3 θ e e + y = e x y = 3 V > ω > x = x = y V () = V sin(ω) f(x) = /(x + ) g(x) = f(x) f [, b] g [, b] g() = 5 g(/).5.5 f() = 3 f() f(3 ) g( ) f( ) y = cos θ sin θ θ = θ = π x(x + ) sin θ cos θ θ 8 9 : ; < = sin θ = x + = cos θ sin θ cos θ θ sin(θ) = sin θ cos θ cos(θ) = cos θ sin θ. sin θ cos θ θ = θ sin θ cos θ θ y = an x +, y() =. I m,n = xm ( x) n m, n I m,n = I n,m f() f() T T 6 7 P P = 6.e. r() = 5e. = = 6?

10 h 7% h = e gk + e gk h 9. 9 R = R e.5 h Q() I() I() = I e Q Q() I() Q () = I(). m g v = mg k h v ( e k/m ), k + r() h = 5 = g v = k (e gk e gk e gk + e gk ). v e. r() = = 5 r() = Q() = A B C? = D > E F C B? H xe x e x xe x xe x (xex ) = (x)ex + x (ex ) = e x + xe x. xe x e x (xex e x ) = (xex ) (ex ) = e x + xe x e x = xe x. xe x = xe x e x + C e x xe x

11 m / ž : : ' ( \ I X P b \ I X P b X ž q. C VCŠ ) l ) +. j / u 3 (/ Z e ) ö j e.) œ C C x_ ) + m.iu. u T. Ai š C C. q 3=.. q Z # Z ) ) ; ž g ž ^ ž ž hc # # }.C # } A. ˆ ) I ž P \ W Y J N : 333 # % lo 3 # ' ( ) * C T. + C., - #. ( ) / / C ) C ) 9 9 J \X \ J WHX J \B\ \ J WHX q% ' q% ' q% a x a. % 9 (# (# x(# ;q% ~(# % : ; : : ; : : : : : : )} )HˆŠ + 3;T. C a C /i ( C O (. >.. m o (. #. #. (# x x ) qj ) DC CŠ C DCŠ j )# j xkc C j. q. / ) ; ) ) u / H q.( +/ q ( +/ q.( CŠ/ q ( CŠ/ q ( # aoi.( aoi ( q ( CŠ/ u CŠ/ A 6) 9 ;q 9 < j 9 < j 9 < 9 =?? + =? = ) ) 5 > 3;T. > ;qu ~ > e \ W Y J N # #

12 - 3 Ž C Ž ž F ( ( Ž Ž Ž Ž ž 3. / Xe J Y B \ / i j i j i _ + ; / # C ;@C I ž PVb - - qd_ /. ) + 3 %. T 3 ) + Dx ) + % g g C š ( HˆŠ} > ( } ˆŠ} ~ I C ˆ I C (q. Š I T ( ( } 333 ;@i ;@i C MCŠhCŠ C } } 3 r ˆ 33 ˆ j x p. p p x C VC C VCŠCŠ q m ( CŠ(q DC Hul.(q DC _/Hˆ q. ; H (q. ; j ai ai j aix m j j.. T 3 x Hˆ 3;Tnmmm. 3;T i m 3;T.B A < < + < < A ( A W Hˆ V # x j e j e B * ) ) ( -/? (? ; % % ; % q% %O %O ; % x. T B ( C ) % Hˆ ( Hˆ E u 3 ˆ i..ai.. aiu.. ai.. # q.. x e aio j eaio x. q. ul. aiul. ~ q m + ~ H m ) H m aoi aoiou~ + D E D F D D H D I F E # uˆ e j + x + # # Ž ˆ # # ˆ # # / (# F F _ F # Z # F H ( iu.( m i F I J E T 3 J F m o 5 + T 3 J J H 3;T. m o J I u F K L M N K O M, K P M E Q R T V E ;q. ;q. Z >/; /; > ;qu ~ ;qu /~ ; / H ; _ I (#. Z E E E F j E / E H. / E I / D E D F K L M W X V Y A K O M Z D K L M D H i D I i i F E i/ F F CŠ F #. F H m F I _/ J E ; J F ai +/ >/ J q j J H.. J I H; E ~ F. u io H I # Z Q E / / Z Q F K L M # ) % ; % K O M Q K L M # ; K O M K P M Q H K L M F ;q% ~ q% % K O M E K L M [ \ ] X ^ X V _ ` L X K O M a R X ^ X V _ ` L X Q Q ž ;q

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