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 ( +).
( ) +) = +) = 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 )
(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.
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.
e + e + e e = + e = e e + e = + e e = = ln + C = ln + e + C = ln( + e ) + C. ' ( ) * +, ( + e ) ) -. /. - 3 - ) ) 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 ).
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
(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 = (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 +
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()
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?
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
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