.
. , , . . [2, 12, 17].
1.1. cosx sin2x dx
►. , dsinx = cosx
. sin2x dsinx=(sin3x)/3+c. ◄
1.2.
►
◄
2.1. x2xdx
► .
uv`dx = uv-. vu`dx)
u = x,v`=2x. :
. x2xdx = x2xlog2e - 2xlog2e dx = x2xlog2e 2x(log2e)2 +c◄
2.2. arccos2xdx
► u = arccos2x, dv = dx,
du = , v = x.
. arccos2xdx = xarccos2x+2. ,
, :
. arccos2xdx = xarccos2x 2 ◄
2.3. .
► u = cosx, v`= ex
:
I :
. I = excosx + exsinx I
I I
. ◄
2.4.
►
,
, . , .
◄
3. :
3.1. .
►
,
4x2 8x = A(x 1)(x2 + 1)2 + B(x2 + 1) 2 + (Cx + D)(x 1)2(x2 + 1) + (Ex +F)(x 1)2
, . . x = 1, B = 1. x = i, :
-4 8i = (Ei + F)(i 1)2 = 2E 2iF.
,
4 = 2E,-8 = 2F, .. E = 2, F = 4.
( ), , x = 1.
8x 8 = A(x2 + 1)2 + 2B(x2 + 1)2x+
= 1 0 = 4A + 8B, .. A = 2. , , x = i.
|
|
8x 8 = (Cx + D)(x 1)22x + E(x 1)2 + (Ex + F)2(x 1) +
x=i, 2 :
C = 2, D = 1. ,
.
,
.
.◄
.
4.1.. I=
► t = tg(x/2), (2n 1) p < x < (2n + 1)p (n = 0;1; 2;), :
:
I(2pn+p-0)=I(2pn+p+0),
= 0 .
2pn < x + p < (2n + 2)p; n < (x + p)/2p < n+1 ,
. ,
I=
I= ◄