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1  sin 2t  sin 2t   1
 
    dt  dt  1.


0  sin 2t   sin 2t  sin 2t   sin 2t   0
1  1 dxex  x 1  1x   2 e x dx 1 e x dx 1 e x dx
10.27  3   3   2 2  3 
0  1x  0  1x  0  1x  0  1x 
1
1 e x dx 1  1  1 e x dx e x 1 e x dx e


  2  x  d  2   2  2  2   . 1
 e

0  x  1 0   x  1  0  x  1  x  1 0 0  x  1 4
1 xe x dx 1  1x 1  dxe x 1 e x dx 1 e x dx
10.28  2   2     2 
0  1x  0  1x  0 x 1 0  1x 
1
1 e x dx 1 1 e x dx e x 1 e x dx e
 d
   e x  1         1.


0 x  1 0  x 1 0 x  1 x  1 0 0 x  1 2
 2  2 2013
dx sin x dx
10.29 I  2013   2013 2013 
0 1  ctg x 0 sin x  cos x
заміна  2 cos 2013 x dx
   . Тоді
x   2  t cos 2013 x  sin 2013 x
0
1   2 sin 2013 x dx  2 cos 2013 x dx  1  2 
 
I        dx  .
2 sin 2013 x  cos 2013 x cos 2013 x  sin 2013 x  2 4
 0 0  0
10.30
 2  2  2
cosx sin x 2 cos  x  4
 ln ctg x 1 dx   ln dx   ln dx 
 4  4 sin x  4 sin x
 2  2  2
 ln 2
  ln 2 dx   ln cos  x   4 dx   ln sin dxx  ,
 4  4  4 8
 2  2
3

оскільки ln cos  x  4 dx  x  t   ln sin dtt .
 4 4  4
x  tg t , t  arctg x
1
ln  1x 
10.31  2 dx  dx 
0 1  x dt  2
1  x
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