Biblioteczka Opracowań Matematycznych

1    _f dz    1    „1    u    _ 1^L    tgx    _

— [—-= — arctgz + C = — arctg — + C = — arctg-+ C

2    ^Jz² + 1    2    2    2    2    2

^ j"2 cos 2x cos 3 xdx = — J (cos av + cos bx\tx = /

a + b

2

a-b

= 2

= 3

<3 = 5 b = -1

10

a + b

2

a-b

= 2

= 5

a = 7 b = -3

<• dx rsinx ^ r sinx*fr ■'sinx •'sin²x •'l—*

’l-cos²x

iln|z-l|-iln|z + l| ₊ C = iln

COS X -1

cosx + l

+c

fsin⁵jr    fSin*xsinxdx f(l-cos²jt)²siaxdx |cosjr=f | _f(l-r)fdi

Icos⁵*    cos’jr    cos³* "|-sin*fr=<*|^_    /’

130/ i- dx _ r */r    _ — +x=l _ r dl _ i- dl r dl 1 r dl

w^-r7^n". . ~w^_J—. i_c

■'cost ^J •/«    .    ,    .

sin +xl (£t=<A

V2

2sin cos 2 2

1 ,

131/

dx

+ C

4+(rf)-

cos²JC+sin²;t . r cos² xdx

+ C

sin _

2 -2L_2

i

cos

r sin²xt/r r r

I--r-= J—⁺ I’

•'cinrrnc r •^łcinv ^Jt

s\nx

sm.tcos² x /sinArcos²* ■'sin* ^J cos² x

f «    = r

^Jsinarcos²x ^J

=^^lnM^-)^{+c =} ^^lnM-

132/    jVg²xdEr = J——| J<& = /gx-x + C

Wykorzystano wzór:

(1.42)    .    ,

sin ,v cos x

^J CII

1

dr =

+ C

cosx

<?* = ₂

COS X

133/

r .t<ir

-I

du-dx v = lgx

= xtgx - \igxdx = Ar/zr - f^Sm*<£ = xtgx + Inlcos jc| + C ^J    JCOSX

134/

I

5sinr+cosc

5sinx cost - +

2sirf xcost+2coś x tgx = t

dc

_ r 5sirtt+cost cojjt^ f cost cost <& _ r 5fgx+i dx •'2sirf.rcost+2cośjt I 1 2sirfjr₊₁ co^x ' 2i£x+2 cos x co ix    coix

2i + -

dc

= dt

1    c5m-3    5    ⁺ ^    5 _f^aTt    5 ₍2tdt    6 t dl

= - [±±id,₌l f-5_-dt = - \—^di = - f4=-+- \-r— =

2    11² + l    4/    /² + l    4^Jr^J + l    4 lf² +1    4^Jr^J+l

5    3    5    1 3

= — ln|/² +1| + — arctgt + C = — ln|/g²x +1| + - arctg (tgx)+ C