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Консультация № 177736
Обсуждение
Неизвестный
08.04.2010, 19:36
общий
это ответ
Здравствуйте, Venya.
1. [$8747$](1/(arcSin2x)[$8730$](1-x2)dx = [$8747$](1/(arcSin2x)d(arcSin(x)) = -1/(arcSin(x))
2. [$8747$](x2+2x+3)Cos2xdx = [u=x2+2x+3 => du = (2x+2)dx; dv = Cos2xdx => v=1/2Sin2x] = 1/2(x2+2x+3)Sin2x - 1/2[$8747$](2x+2)Sin2xdx = [u=2x+2 => du=2dx; dv=Sin2xdx => v=-1/2Cos2x] = 1/2(x2+2x+3)Sin2x - 1/2(-(2x+2)Cos2x + 1/2[$8747$]2Cos2xdx) = 1/2(x2+2x+3)Sin2x + (x+1)Cos2x -1/4Sin2x
1. [$8747$](1/(arcSin2x)[$8730$](1-x2)dx = [$8747$](1/(arcSin2x)d(arcSin(x)) = -1/(arcSin(x))
2. [$8747$](x2+2x+3)Cos2xdx = [u=x2+2x+3 => du = (2x+2)dx; dv = Cos2xdx => v=1/2Sin2x] = 1/2(x2+2x+3)Sin2x - 1/2[$8747$](2x+2)Sin2xdx = [u=2x+2 => du=2dx; dv=Sin2xdx => v=-1/2Cos2x] = 1/2(x2+2x+3)Sin2x - 1/2(-(2x+2)Cos2x + 1/2[$8747$]2Cos2xdx) = 1/2(x2+2x+3)Sin2x + (x+1)Cos2x -1/4Sin2x
Неизвестный
09.04.2010, 00:06
общий
это ответ
Здравствуйте, Venya.
3. [$8747$]xdx/((x2+4*x+5)*(x+2))
x/((x2+4*x+5)*(x+2))=(2*x+5)/(x2+4*x+5)-2/(x+2)
[$8747$]xdx/((x2+4*x+5)*(x+2))=[$8747$](2*x+5)dx/(x2+4*x+5)-2*[$8747$]dx/(x+2)=[$8747$]2*(x+2)d(x+2)/(x2+4*x+5)+[$8747$]d(x+2)/(x2+4*x+5)-2*[$8747$]d(x+2)/(x+2)=[$8747$]d(x2+4*x+5)/(x2+4*x+5)+[$8747$]d(x+2)/((x+2)2+1)-2*[$8747$]d(x+2)/(x+2)=ln(x2+4*x+5)+arctg(x+2)-2*ln(x+2)
4. [$8747$]sin(x)dx/(5+8*cos(2*x))
cos(2*x)=2*cos2(x)-1
5+8*cos(2*x)=5+16*cos2(x)-8=(4*cos(x))2-([$8730$]3)2)
sin(x)dx=-(1/4)*d(4*cos(x))
[$8747$]sin(x)dx/(5+8*cos(2*x))=-(1/4)*[$8747$]d(4*cos(x))/(4*cos(x))2-([$8730$]3)2)=(-1/4)*(-1/[$8730$]3)*arctg(cos(x)*4/[$8730$]3)=[$8730$]3/12*arctg((4*[$8730$]3/3)*cos(x))
3. [$8747$]xdx/((x2+4*x+5)*(x+2))
x/((x2+4*x+5)*(x+2))=(2*x+5)/(x2+4*x+5)-2/(x+2)
[$8747$]xdx/((x2+4*x+5)*(x+2))=[$8747$](2*x+5)dx/(x2+4*x+5)-2*[$8747$]dx/(x+2)=[$8747$]2*(x+2)d(x+2)/(x2+4*x+5)+[$8747$]d(x+2)/(x2+4*x+5)-2*[$8747$]d(x+2)/(x+2)=[$8747$]d(x2+4*x+5)/(x2+4*x+5)+[$8747$]d(x+2)/((x+2)2+1)-2*[$8747$]d(x+2)/(x+2)=ln(x2+4*x+5)+arctg(x+2)-2*ln(x+2)
4. [$8747$]sin(x)dx/(5+8*cos(2*x))
cos(2*x)=2*cos2(x)-1
5+8*cos(2*x)=5+16*cos2(x)-8=(4*cos(x))2-([$8730$]3)2)
sin(x)dx=-(1/4)*d(4*cos(x))
[$8747$]sin(x)dx/(5+8*cos(2*x))=-(1/4)*[$8747$]d(4*cos(x))/(4*cos(x))2-([$8730$]3)2)=(-1/4)*(-1/[$8730$]3)*arctg(cos(x)*4/[$8730$]3)=[$8730$]3/12*arctg((4*[$8730$]3/3)*cos(x))
09.04.2010, 13:43
общий
это ответ
Здравствуйте, Venya.
4. Замена u=cos x
∫sin(x)dx/(5+8cos(2x))=-∫d(cos(x))/(5+8(2cos2x-1))= [$8747$]du/(3-16u2)=
=(1/16)[$8747$]du/((3/16)-u2)=(1/16)(4/(2[$8730$]3)ln((|[$8730$]3/4)+u|/|([$8730$]3/4)-u|)+C=
=(1/(8[$8730$]3))ln(|[$8730$]3+4u|/|[$8730$]3-4u|)+C=
=(1/(8[$8730$]3))ln(|[$8730$]3+4cos x|/|[$8730$]3-4cos x|)+C
4. Замена u=cos x
∫sin(x)dx/(5+8cos(2x))=-∫d(cos(x))/(5+8(2cos2x-1))= [$8747$]du/(3-16u2)=
=(1/16)[$8747$]du/((3/16)-u2)=(1/16)(4/(2[$8730$]3)ln((|[$8730$]3/4)+u|/|([$8730$]3/4)-u|)+C=
=(1/(8[$8730$]3))ln(|[$8730$]3+4u|/|[$8730$]3-4u|)+C=
=(1/(8[$8730$]3))ln(|[$8730$]3+4cos x|/|[$8730$]3-4cos x|)+C
Форма ответа
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