, 22 2011 . 18:07
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y' y'=(4x-2)y/x 2 +2√y ( ) 2√y z=√y z'=(2x-1)z/x 2 +2 ( ) z'=(2x-1)z/x 2 dz/z=(2x-1)dx/x 2 ln|z|=2ln|x|+(1/x)+const z=Cx 2 e 1/x : z=C(x)x 2 e 1/x , C'(x)=(2/x 2 )e -1/x C(x)=2e -1/x +C z=2x 2 +Cx 2 e 1/x : y=(2x 2 +Cx 2 e 1/x ) 2 , y=0 (y=0 2√y)
, 24 2011 . 08:41
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? : |a n |=1/(n4 n ) : 1)|a n |=1/(n4 n ), |a n+1 |=1/((n+1)4 n+1 ) n+1>n 4 n+1 >4 n , (n+1)4 n+1 >n4 n 1/((n+1)4 n+1 )<1/(n4 n ) 2)|a n |=1/(n4 n )≤1/n ----> 0 n--->∞. , |a n | ----> 0. , .
, 20 2011 . 10:53
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: ) y=1-lnSin(x^2-1) 3 ≤x ≤4 ) x=2(2Cost-Cos2t) y=2(2Sint-sin2t) 0 ≤ t ≤/3 ) ρ=8(1-Cosφ) -2π/3 ≤φ≤0 ). x = 2 (2 cos t - cos 2t), y = 2 (2 sin t - sin 2t). x'(t) = 2 (-2 sin t + 2 sin 2t) = 4 (sin 2t - sin t) = 4 2 cos (3t/2) sin (t/2) = 8 cos (3t/2) sin (t/2) , y'(t) = 2 (2 cos t - 2 cos 2t) = 4 (cos t - cos 2t) = 4 2 sin (3t/2) sin (t/2) = 8 sin (3t/2) sin (t/2), (x'(t)) 2 = 64 cos 2 (3t/2) sin 2 (t/2), (y'(t)) 2 = 64 sin 2 (3t/2) sin 2 (t/2), (x'(t)) 2 + (y'(t)) 2 = 64 cos 2 (3t/2) sin 2 (t/2)+ 64 sin 2 (3t/2) sin 2 (t/2) = 64 (sin 2 (3t/2)+ cos 2 (3t/2)) sin 2 (t/2) = = 64 9; sin 2 (t/2). : dL = √((x'(t)) 2 + (y'(t)) 2 ) dt = √(64 sin 2 (t/2)) dt = 8 sin (t/2) dt. ∫sin (t/2) dt = -2 cos (t/2) + C. : L = 0 ∫ π/3 (8 sin (t/2)) dt = -8 cos (t/2)| 0 π/3 = -8 (cos (π/6) - cos 0) = -8 ((√3)/2 - 1) =8 - 4√3 ≈ 1,072. : 8 - 4√3 ≈ 1,072. ). ρ = 8 (1 - cos φ). ρ' = 8 (1 - cos φ)' = 8 sin φ, ρ 2 = 64 (1 - 2 cos φ+ cos 2 φ), (ρ') 2 = 64 sin 2 φ, ρ 2 + (ρ') 2 = 64 (1 - 2 cos φ + cos 2 φ) + 64 sin 2 φ = 64 (1 - 2 cos φ) + 64 (cos 2 φ +sin 2 φ) = 64 (2 - 2 cos φ) = = 64 (2 - 2 (2 cos 2 (φ/2) - 1)) = 64 (2 - 4 cos 2 (φ/2) + 2) = 64 (4 - 4 cos 2 (φ/2) = 256 (1 - cos 2 (φ/2)) =256 sin 2 (φ/2), √(ρ 2 + (ρ') 2 ) = √(256 sin 2 (φ/2)) = 16 sin (φ/2), ∫sin (φ/2) dφ = ∫sin (φ/2) d(2φ/2) = 2 ∫sin (φ/2) d(φ/2) = -2 cos (φ/2) + C. L : L = -2π/3 ∫ 0 (16 sin (φ/2)) dφ = 4π/3 ∫ 2π (16 sin (` 6;/2)) dφ =-32 cos (φ/2)| 4π/3 2π = = -32 (cos π - cos (2π/3)) = -32 (-1 - (-0,5)) = -32 (-0,5) = 16. : 16.
, 12 2010 . 06:50
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lnA=1/z*ln(1-cos5/z) =>Lim(z->∞)lnA= Lim(z->∞)[ln(1-cos5/z)/z]=∞/∞=ln Lim(z->∞)[(1-cos5/z)/z]= ln Lim(z->∞)[(-5sin5/z)/z2]=0/∞=-5sin5/z)/z2]=0/∞=-5ln Lim(z->∞)[(-5cos5/z)/z3]=25ln Lim(z->∞)[(cos5/z)/z3]=0 => lnlimA=0 =>limA=e0=1: lim(x->0)(1-cos5x)x=1
, 25 2010 . 23:33
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f() :f(x) = ( - 3)^2 (0,3).: 2*L:f(x)=a0/2+∑n=1∞(an*cos(Pi*n*x/L)+bn*sin(Pi*n*x/L))a0=(1/L)*∫03f(x)dx=(2/3)*∫(x-3)2dx=(2/3)*(x-3)3/3|03=6 :∫x*cos(a*x)dx=(1/a)*(x*sin(a*x)-∫sin(a*x)dx)=(1/a)*(x*sin(a*x)+cos(a*x)/a)+C∫x*sin(a*x)dx=(1/a)*(-x*cos(a*x)+∫cos(a*x)dx)=(1/a)*(-x*cos(a*x)+sin(a*x)/a)+C∫x2*cos(a*x)dx=(1/a)*(x2*sin(a*x)-∫2*x*sin(a*x)dx)=(a2*x2*sin(a*x)-2*sin(a*x)+2*a*x*cos(a*x))/a3+C∫x2*sin(a*x)dx=(-a2*x2*cos(a*x)+2*cos(a*x)+2*a*x*sin(a*x))/a3+CC , sin(Pi*n)=0, cos(Pi*n)=(-1)n, n=0,1,2,3... :an=(2/3)*& #8747;03(x-3)2*cos(Pi*n*x*2/3)dx=(2/3)*∫03(x^2-6*x+9)*cos(Pi*n*x*2/3)dx=(2/3)*∫03x2*cos(Pi*n*x*2/3)dx-4*∫03x*cos(Pi*n*x*2/3)dx+6*∫03cos(Pi*n*x*2/3)dx=(9*(Pi*n-sin(Pi*n)*cos(Pi*n)))/(Pi3*n3)=9/(Pi2*n2)bn=(2/3)*∫03(x-3)2*sin(Pi*n*x*2/3)dx=(2/3)*∫03(x^2-6*x+9)*sin(Pi*n*x*2/3)dx=(2/3)*∫03x2*sin(Pi*n*x*2/3)dx-4*∫03x*sin(Pi*n*x*2/3)dx+6*∫03sin(Pi*n*x*2/3)dx=(9*(-1+Pi2*n2+cos(Pi*n)2))/(Pi3*n3)=9/(Pi*n)f(x)=3+∑n=1∞[9*cos(2*Pi*n*x/3)/(Pi2*n2)+9*sin(2*Pi*n*x/3)/(Pi*n)] f(x)( ) () n=5.2 0,001 :∫(sin(x^2)/x)dx 0 0,5. sin(t)=∑k=0∞(-1)k*t2*k+1/(2*k+1)!t=x2sin(x2)=∑k=0∞(-1)k*t4*k+2/(2*k+1)!sin(x2)/x=∑k=0∞(-1)k*t4*k+1/(2*k+1)!∫01/2(sin(x2)/x)dx=∫01/2(∑k=0∞(-1)k*x4*k+1/(2*k+1)!)dx=∑k=0∞(-1)k/(2*k+1)! *∫01/2x4*k+1dx=∑k=0∞(-1)k/((2*k+1)!*24*k+2*(4*k+2)) n=2∫01/2(sin(x2)/x)dx=∑k=02(-1)k/((2*k+1)!*24*k+2*(4*k+2))=1/8-1/2304+1/1228800=459203/3686400 =0,1245668...3: ∫(3x-y)+i(x+3y)dz , , (0, 0), (3, 3) (0, 6)..∫Lf(z)dz=∫Ludx-vdy+i*∫Lvdx+udyu=3*x-yv=x+3*y∫Cf(z)dz=∫OAudx-vdy+∫ABudx-vdy+i*(∫OAvdx+udy+∫ABvdx+udy) OA: y=x, dy=dx, 0≤ x ≤ 3∫OAudx-vdy=∫OA(3*x-y)dx-(x+3*y)dy=∫03(-2*x)dx= -9∫OAvdx+udy=∫OA(x+3*y)dx+(3*x-y)dy=∫036*xdx=27 AB: y=6-x, dy= -dx, 3≤ x ≤ 6∫ABudx-vdy=∫AB(3*x-y)dx-(x+3*y)dy=∫36(2*x+12)dx=63∫ABvdx+udy=∫AB(x+3*y)dx+(3*x-y)dy=∫36(-6*x+24)dx= -9∫Cf(z)dz= (-9+63)+i(27+(-9))=54+18*i
, 14 2010 . 18:25
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" ": , : r=cos2φ. . :S=1/2αβ∫r2(φ)dφ.. 0 S=2S1S=0∫cos2φdφ=0∫(1+cos4φ)/2dφ=1/2φ/0+1/8*sin4φ/0=/2. " ": , :x=2*((sint)^2), =0, y>0 , y=3cost t Pi/2 0S=∫abx(t)*y'(t)dty'(t)=-3*sin(t)S=∫Pi/20x(t)*y'(t)dt=∫Pi/202*sin2(t)*(-3*sin(t))dt=6*∫0Pi/2sin3(t)dt=6*∫0Pi/2(1-cos2(t))*sin(t)dt=6*(-cos(t)+cos3(t)/3)|0Pi/2=6*(1-1/3)=4. " ": , :y=1/((x*x+1)^2)x>=0y=0. :S=∫0∞dx/(x2+1)2=∫0∞(x2x2)/(x2+1)2)dx=∫0∞dx/(x2+1)-∫0∞x2dx/(x2+1)2 :∫udv=u*v-∫vduu=xdv=xdx/(x2+1)2v=∫xdx/(x2+1)2=(1/2)*∫(x2+1)-2d(x2+1)=-(1/2)*1/(x2+1)∫udv=∫x2dx/(x2+1)2=-(1/2)*x/(x2+1)+(1/2)*∫dx/(x2+1)=-(1/2)*x/(x2+1)+(1/2)*arctg(x)∫dx/(x2+1)2=arctg(x)-(-(1/2)*x/(x2+1)+(1/2)*arctg(x))=1/2*(arctg(x)+x/(x2+1))S=∫0∞dx/(x2+1)2=1/2*(arctg(x)+x/(x2+1))|0∞=(1/2)*limx->∞(arctg(x)+x/(x2+1))=(1/2)*limx->∞arctg(x)+(1/2)*limx->∞x/(x2+1)=(1/2)*Pi/2+(1/2)*0=Pi/4
, 10 2010 . 11:08
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Lim ∫c(x^n)d(x^n) n ∞ [0,1].((x^n) - , x^n) , . : xn= t, 0 ≤ x ≤ 1. 0 ≤ t ≤ 1, 0∫1c(xn)d(xn) =0∫1f(t)dt = F(t)|01= F(1) F(0) = C(1) C(0) (C c(xn)). (y sin2x) ∙ cos x = y ∙ sin x. . :y sin2x = y ∙ tg x, (cos x ≠ 0)y y ∙ tg x = sin2x. (1) y + p(x)y = g(x), . (1) . y = u(x)v(x). y = uv + uv. (1) uv + uv uv ∙ tg x = sin2x,uv + u(v v ∙ tg x) = sin2x. (2) v(x) , . v v ∙ tg x = 0:dv/dx v ∙ tg x = 0,dv/dx = v ∙ tg x,dv/v = tg x ∙ dx,ln |v| = -ln |cos x| + ln |C|. , C = 1. v = 1/cos x. v = 1/cos (2), u'/cos x = sin2x,du/dx = sin2x ∙ cos x,du = sin2x ∙ cos x ∙ dx,∫du = ∫sin2x ∙ cos x ∙ dx,∫du = ∫sin2x ∙ d(sin x),u = (sin3x)/3 + C., y = uv = ((sin3x)/3 + C)/cos x .: y = ((sin3x)/3 + C)/cos x.
