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Реферат: Решения к Сборнику заданий по высшей математике Кузнецова Л.А. - 2. Дифференцирование. Зад.3

Задача 3 . Найти дифференциал .

3.1.

dy= arcsin(1/x)dx-x/√(1-1/x2 )* dx/x2 +((1+x/√(x2 -1))/(x+√(x2 -1)))dx= arcsin(1/x)dx-dx/√(x2 -1)+ ((x+√(x2 -1))/ ((x+√(x2 -1))√(x2 -1)))dx= arcsin(1/x)dx-dx/√(x2 -1)+ dx/√(x2 -1)= arcsin(1/x)dx

3.2.

dy= dx/cos2(2arccos√(1-2x2 ))*(-2/√(1-√(1-2x2 )))*(-2x/√(1-2x2 ))= 4xdx/ (cos2(2arccos√(1-2x2 ))*√ (1-2x2 -√(1-2x2 )))

3.3.

dy= dx/√(1+2x)-((1+1/√(2x+1))/(x+√(1+2x))))dx= dx/√(1+2x)-((√(2x+1)+1)/(√(2x+1)*(x+√(2x+1))))dx= ((x+√(2x+1)- √(2x+1)-1)/( √(2x+1)*(x+√(2x+1))))dx= ((x-1)/(x+√(2x+1)))dx

3.4.

dy=2xarctg√(x2 -1)dx-x2 dx/(1+x2 -1)-xdx/√(x2 -1)= 2xarctg√(x2 -1)dx-dx-xdx/√(x2 -1)

3.5.

dy= dx/√(1-1/(1+2x2))*4x/2√(1+2x2 )3 = 2xdx/√(2x2 (1+2x2 )3 /(1+2x2 ))= 2xdx/((1+2x2 )√( 2x2 ))= √2dx/(1+2x2 )

3.6.

dy= ln│x+√(x2 +3)│dx+xdx/(x+√(x2 +3))*(1+x/√(x2 +3))= ln│x+√(x2 +3)│dx+ xdx/(x+√(x2 +3))*(x+√(x2 +3))/√(x2 +3)= ln│x+√(x2 +3)│dx+ xdx/√(x2 +3)

3.7.

dy= (сhx/(1+sh2 x)+сhxlnchx+sh2 x/chx)dx

3.8.

dy= ((-1/√(1-(x2 -1)2 /2x4 ))*(2√2x3 -2√2x3 +2√2x)/2x4 )dx= -2√2xdx/(√2x2 √(x4 +2x2 -1))= 2dx/(x√(x4 +2x2 -1))

3.9.

dy=((-2cosxsinx-(4cos3 xsinx)/(2√(1+cos4 x)))/(cos2 x+√(1+cos4 x)))dx=

((-sin2x*√(1+cos4 x)-sin2x*cos2 x)/(cos2 x*√(1+cos4 x)+1+cos4 x))dx

3.10.

dy=((1+x/√(1+x2 ))/(x+√(1+x2 ))-xarctgx/√(1+x2 )- √(1+x2 )/ (1+x2 ))dx=

(1/√(1+x2 )-xarctgx/√(1+x2 )-1/√(1+x2 ))dx= -xarctgxdx/√(1+x2 )

3.11. .

dy=((1+x2 -2x2 lnx)/(x(1+x2 ))-(( 1+x2 )/2x2 )*((2x(1+x2 )-2x3 )/( 1+x2 )2 ))dx=

((x+x3 -2x3 lnx)/(x(1+x2 )2 )-(( 1+x2 )x)/(x2 (1+x2 )2 ))dx=

((x+x3 -2x3 lnx-x-x3 )/(x(1+x2 )2 )dx= -2xlnxdx/(1+x2 )2

3.12.

dy=((ex + e2x /√( e2x -1))/( ex +√( e2x -1))+ex /√(1-e2x ))dx=

(ex (ex +√( e2x -1))/((ex +√( e2x -1))√( e2x -1))+ ex /√(1-e2x ))dx=

(ex /√(e2x -1)+ex /√(1-e2x ))dx

3.13.

dy=(√(4-x2 )-2x2 /(2√(4-x2 ))+a/(2√(1-x2 )))dx=((4-3x2 )/√(4-x2 )+a/(2√(1-x2 )))dx

3.14.

dy=(1/(2tg(x/2)cos2 (x/2))-(sinx-xcosx)/sin2 x)dx=(1/(1-cosx)-(sinx-xcosx)/((1-cosx)(1+cosx)))dx=((1+cosx-sinx+xcosx)/(1-cos2 x))dx

3.15.

dy=(2+(cosx-2sinx)/(sinx+2cosx))dx

3.16.

dy=(-1/(2√(ctgx)sin2 x)-2tg2 x/(6√(tg3 x)cos2 x))dx=((-cos4 x*√(tg3 x)-sin4 x*√(ctgx))/(4cos4 x*sin2 x*√(ctgx*tg3 x)))dx=((-cos4 x*√(tg3 x)-sin4 x*√(ctgx))/(4cos3 x*sin3 x))dx=((-cos4 x*tg2 x-sin4 x)/(4cos3 x*sin3 x*√(tgx)))dx=((-cos2 x*sin2 x-sin4 x)/(4cos3 x*sin3 x*√tgx))dx=((-cos2 x-sin2 x)/(4cos3 x*sinx*√tgx))dx=((-√ctgx)/(4cos3 x*sinx))dx

3.17.

dy=((x/(x+√(x2 +1)))*((2x(1+x/√(x2 +1)-2(x+√(x2 +1))))/(4x2 )))dx=((x/(x+√(x2 +1)))*((x√(x2 +1)+x2 -x√(x2 +1)-x2 -1)/x2 ))dx=-dx/(x2 +x√(x2 +1))

3.18.

dy=(1/3*3 √((x-2)/(x+2))2 *(x-2-x-2)(x-2)2 )dx=(-4/(3(x-2)2 )*3√((x-2)/(x+2))2 )dx

3.19.

dy=((2x2 -x2 +1)/(x2 (1+(x2 -1)2 /x2 )))dx=((x2 (x2 +1))/(x2 (x2 +(x2 -1)2 )))dx=((x2 +1)/(x4 -x2 +1))dx

3.20.

dy=(2x/(x2 -1)+2x/(x2 -1)2 )dx=((2x3 -2x+2x)/(x2 -1)2 )dx=(2x3 /(x2 -1)2 )dx

3.21.

dy=(1/((1+(tg(x/2)+1)2 )*(2cos2 (x/2))))dx=(1/((1+tg2 (x/2)+2tg(x/2)+1)*(2cos2 (x/2))))dx=(1/(2(1+2sin(x/2)*cos(x/2)+1)))dx=dx/(4+2sinx)

3.22.

dy=((2+(2x+1)/√(x2 +x))/(2x+2√(x2 +x)+1))dx=((2√(x2 +x)+2x+1)/(√(x2 +x)*(2x+2√(x2 +x)+))dx=dx/√(x2 +x)

3.23.

dy=((-sin√x)/(2√xcos√x)+(tg√x)/(2√x)+√x/(2√xcos2 √x))dx=((-sin√x)/(2√xcos√x)+(sin√x)/(2√xcos√x)+1/(2cos2 √x))dx=((1+tg2 x)/2)dx

3.24.

dy=(ex (cos2x+2sin2x)+ex (-2sin2x+4cos2x))dx=ex (cos2x+2sin2x-2sin2x+4cos2x)dx=5ex cos2xdx

3.25.

dy=((sinlnx-coslnx)+x((coslnx)/x+(sinlnx)/x))dx=(sinlnx-coslnx+coslnx+sinlnx)dx=2sinlnxdx

3.26.

dy=((e2√(x-1) /(2√(x-1)))*(1/√(x-1))+(√(x-1)-1/2)*e2√(x-1) *1/√(x-1))dx=(e2√(x-1) *(1/(2x-2)+1-1/(2√(x-1))))dx=(e2√(x-1) *((2x-1-√(x-1))/2x-2))dx

3.27.

dy=(-sinxlntgx+(cosx/tgx)*1/cos2 x-1/(2tg(x/2)*cos2 (x/2)))dx=(-sinxlntgx+cos2 x/sinx-(1+tg2 (x/2))/2tg(x/2))dx

3.28.

dy=(x/√(3+x2 )-ln│x+√(3+x2 )│-(x(1+x/√(3+x2 )))/(x+√(3+x2 )))dx=(x/√(3+x2 )-ln│x+√(3+x2 )│-(x(√(3+x2 )+x))/((x+√(3+x2 ))√(3+x2 ))dx=(x/√(3+x2 )-ln│x+√(3+x2 )│-x/√(3+x2 ))dx=-ln│x+√(3+x2 )│dx

3.29.

dy=(1/2√x-arctg√x-(1+x)/((1+x)*2√x))dx=(1/2√x-arctg√x-1/2√x)dx=-arctg√xdx

3.30.

dy=(arctgx+x/(1+x2 )-(2x/√(1+x2 ))*1/(2√(1+x2 )))dx=(arctgx+x/(1+x2 )-x/(1+x2 ))dx=arctgxdx

3.31.

dy=(√(x2 -1)+x/√(x2 -1)+(1+x/√(x2 -1))/(x+√(x2 -1)))dx=(√(x2 -1)+x/√(x2 -1)+(x+√(x2 -1))/(√(x2 -1)(x+√(x2 -1))))dx=(√(x2 -1)+x/√(x2 -1)+1/√(x2 -1))dx=((x2 -1+x+1)/√(x2 -1))dx=(x2 +x)dx/√(x2 -1)