| 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)
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