Propensity Score Matching - Propensity score matching methods for the analysis of recurrent eve

Demostrar las siguientes Integraciones: 1. sen2_{x . dx}

= x - sen 2x + c . 2 4

(½ - ½ cos 2x) dx = ½ dx - ½ . ½cos 2x . dx = x - ¼ sen 2x = x - sen 2x + c . 2 4 2. sen4_{x . dx} = 3x - sen 2x + sen 4x + c . 8 4 32 sen2_{x . sen}2_{x dx} = (½ - ½ cos 2x)2 dx = {(½)2_{- 2(}_½)₍_½)_{cos 2x + [(}_½)_{cos 2x]}2_{} dx} {¼ - ½ cos 2x + ¼ cos2_{2x} dx} ¼ dx - ½cos 2x dx + ¼ cos2_{2x dx} ¼ dx - ½.½cos 2x .(2)dx + ¼ [½ + ½ cos 2(2x)] dx ¼ dx - ¼ cos 2x .(2)dx + ¼ [½ + ½ cos 4x] dx ¼ dx - ¼ cos 2x .(2)dx + ¼.½dx + ¼.½cos 4x dx ¼ dx - ¼ cos 2x .(2)dx + ¼.½dx + ¼.½.¼ cos 4x .(4)dx ¼ dx - ¼ cos 2x .(2)dx + ⅛ dx + 1/32cos 4x .(4)dx ¼ x - ¼ sen 2x + ⅛ x + 1/32 sen 4x = ¼ x + ⅛ x - ¼ sen 2x + 1/32 sen 4x = 2/8 x + ⅛ x - ¼ sen 2x + 1/32 sen 4x = ⅜ x - ¼ sen 2x + 1/32 sen 4x + c . 3. cos4_{x dx} = 3x + sen 2x + sen 4x + c . 8 4 32 cos2_{x. cos}2_{x dx} = [½ + ½ cos 2x]2 dx dx =

{(½)2_{+ 2(}_½)₍_½)_{cos 2x + [(}_½)_{cos 2x]}2_{} dx} {¼ + ½ cos 2x + ¼ cos2_{2x} dx} ¼ dx + ½cos 2x dx + ¼ cos2_{2x dx} ¼ dx + ½.½cos 2x (2)dx + ¼ [½ + ½ cos 2(2)x] dx ¼ dx + ¼ cos 2x (2)dx + ¼ [½ + ½ cos 4x] dx ¼ dx + ¼ cos 2x (2)dx + ¼.½dx + ¼.½cos 4x dx ¼ dx + ¼ cos 2x (2)dx + ⅛ dx + ¼.½.¼ cos 4x (4)dx ¼ dx + ¼ cos 2x (2)dx + ⅛ dx + 1/32cos 4x (4)dx ¼ x + ¼ sen 2x + ⅛ x + 1/32 sen 4x = ¼ x + ⅛ x + ¼ sen 2x + 1/32 sen 4x = 2/8 x + ⅛ x + ¼ sen 2x + 1/32 sen 4x = 3/8 x + ¼ sen 2x + 1/32 sen 4x + c . 4. sen6_{x dx}

= 5x - sen 2x + sen3 2x + 3sen 4x + c . 16 4 48 64

sen2_{x . sen}2_{x . sen}2_{x dx}

= (sen2 x)3dx =

[½ - ½ cos 2x]3_dx

{(½)3 - 3.(½)2. ½ cos 2x + 3(½). (½ cos 2x)2_-_(½_{cos 2x)}3_{} dx}

(⅛ - ⅜cos 2x + 3(½).(¼)cos2_{2x -}_⅛_cos3_{2x) dx}

(⅛ - ⅜cos 2x + ⅜cos2_{2x -}_⅛_cos3_{2x) dx}

{⅛ - ⅜cos 2x + ⅜[½ + ½ cos (2)2x] - ⅛.cos2_{2x. cos 2x} dx}

{⅛ - ⅜ cos 2x + 3/16 + 3/16cos 4x - ⅛.[(1 - sen2_{2x). cos 2x}_{]} dx}

{⅛ + 3/16- ⅜ cos 2x +3/16 cos 4x - ⅛[cos 2x - sen2_{2x.cos 2x}_]}_dx

{2/16 + 3/16 - ⅜ cos 2x + 3/16 cos 4x - ⅛cos 2x + ⅛sen2 2x.cos 2x}dx

{5/16- ⅜ cos 2x - ⅛cos 2x + 3/16 cos 4x + ⅛(sen 2x)2.cos2x}dx

{5/16 - 4/8 cos 2x + 3/16 cos 4x + ⅛(sen 2x)2.cos2x}dx

{5/16 - ½ cos 2x + 3/16 cos 4x + ⅛(sen 2x)2.cos2x}dx

5/16dx - ½ cos 2x .(2)dx + 3/16 cos 4x .(4)dx + ⅛(sen2x)2_{.cos2x dx} 5/16dx - ½.½ cos 2x .(2)dx + 3/16.¼ cos 4x .(4)dx + ⅛.½(sen2x)2_.(2)cos2xdx 5/16dx - ¼ cos 2x .(2)dx + 3/64cos 4x .(4)dx + 1/16(sen2x)2_.(2)cos2xdx 5/16 x - ¼ sen 2x + 3/64 sen 4x + 1/16(sen_2x)3

= 3

5/16 x - ¼ sen 2x + 3/64 sen 4x + 1/48 (sen3_2x) =

5x - sen 2x + sen3_{2x + 3sen 4x + c .} 16 4 48 64

5. cos6_{x dx}

= 5x + sen 2x - sen3 2x + 3sen 4x + c . 16 4 48 64

cos2_{x. cos}2_{x. cos}2_{x dx}

= (cos2 x)3 dx = [½ + ½ cos 2x]3 dx

(⅛ + 3.¼.½ cos 2x + 3(½).(½)2_._cos2_{2x +}_⅛_cos3_{2x) dx}

(⅛ + ⅜cos 2x + 3(½).(¼)cos2_{2x +}_⅛_cos3_{2x) dx}

(⅛ + ⅜cos 2x + ⅜ cos2_{2x +}_⅛_cos3_{2x) dx}

{⅛ + ⅜cos 2x + ⅜[½ + ½ cos (2)2x] + ⅛.cos2_{2x. cos 2x} dx}

{⅛ + ⅜ cos 2x + 3/16 + 3/16cos 4x + ⅛[(1 - sen2_{2x).cos 2x}_{]} dx}

{⅛ + 3/16+ ⅜ cos 2x+3/16cos 4x + ⅛[cos 2x - sen2_{2x.cos 2x}_]}_dx

{2/16 + 3/16 + ⅜cos 2x + 3/16 cos 4x + ⅛cos 2x - ⅛sen2 2x.cos 2x}dx

{5/16 + ⅜ cos 2x + ⅛cos 2x + 3/16 cos 4x - ⅛(sen 2x)2_.cos2x_}_dx

{5/16 + 4/8 cos 2x + 3/16 cos 4x - ⅛(sen 2x)2_.cos2x}dx

{5/16 + ½ cos 2x + 3/16 cos 4x - ⅛(sen 2x)2_.cos2x}dx

5/16dx + ½cos 2x .(2)dx + 3/16cos 4x .(4)dx - ⅛(sen2x)2_{.cos2x dx} 5/16dx + ½.½ cos 2x.(2)dx + 3/16.¼ cos 4x .(4)dx- ⅛.½(sen2x)2_.(2)cos2xdx 5/16dx + ¼ cos 2x .(2)dx + 3/64 cos 4x .(4)dx - 1/16(sen2x)2_.(2)cos2xdx 5/16 x + ¼ sen 2x + 3/64 sen 4x - 1/16(sen 2x)2+1

= 2+1

5/16 x + ¼ sen 2x + 3/64 sen 4x - 1/16 (sen 2x)3 = 3

5x + sen 2x - sen3_{2x + 3sen 4x + c .} 16 4 48 64

6. sen2_{ax dx}

= x - sen 2ax + c . 2 4a .

[½ - ½ cos 2ax]dx = ½ dx - ½ cos 2ax]dx =

½ dx - ½.1/2a cos 2ax . (2a)]dx = ½ x - 1/4a .sen 2ax = x - sen 2ax + c .

2 4a 7. sen2_x/2_._cos2_x/2_dx

= x - sen 2x + c . 8 16

[½ - ½ cos 2(x/2)].[½ + ½ cos 2(x/2)]dx .Simplificando:

[½ - ½ cos x].[½ + ½ cos x]dx. Tenemos una diferencia de cuadrados.

{[½]2 - [½ cos x]2} dx = {[¼] - [¼ cos2 x]} dx = ¼dx - ¼ cos2_{x dx} = ¼dx - ¼

{

[½ + ½ cos 2x] dx

}

¼dx - ¼.½ dx - ¼.½ cos 2x dx = ¼dx - ⅛ dx - ⅛ cos 2x dx ¼dx - ⅛ dx - ⅛.(½) cos 2x .(2) dx = ¼dx - ⅛ dx - 1/16 cos 2x .(2) dx ¼ x - ⅛ x - 1/16 sen 2x = 2/8 x - ⅛ x - 1/16 sen 2x = 1/8 x - 1/16 sen 2x = x - sen 2x + c . 8 16 8. sen4_{ax dx} sen2_{ax . sen}2_{ax dx}

= sen2 ax . sen2 ax dx = [½ - ½ cos 2ax]2dx

{¼ - ½ cos 2ax + ¼ cos2_{2ax} dx}

¼ dx - ½cos 2ax dx + ¼ cos2_{2ax dx}

¼ dx - ½.½acos 2ax (2a)dx + ¼ [½ + ½ cos 2(2ax)] dx ¼ dx - ¼acos 2ax (2a)dx + ¼ [½ + ½ cos 4ax] dx ¼ dx - ¼acos 2ax (2a)dx + ¼.½dx + ¼.½cos 4ax dx ¼ dx - ¼acos 2ax (2a)dx + ¼.½dx + ¼.½.¼acos 4ax

(4a)dx

¼ dx - ¼acos 2ax (2a)dx + ⅛ dx + 1/32acos 4ax (4a)dx

¼ x - ¼a sen 2ax + ⅛ x + 1/32a sen 4ax =

¼ x + ⅛ x - ¼a sen 2ax + 1/32a sen 4ax =

2/8 x + ⅛ x - ¼a sen 2ax + 1/32a sen 4ax =

⅜ x - ¼a sen 2ax + 1/32a sen 4ax =

⅜ x - sen 2ax + sen 4ax + c . 4a 32a . 9. sen2_{2x .cos}4_{2x dx}

sen2_{2x .cos}4_{2x . cos}2_{2x dx}

[½ - ½ cos 2(2x)]. [½ + ½ cos 2(2x)]. cos2_{2x dx}

[½ - ½ cos 4x]. [½ + ½ cos 4x]. cos2_{2x dx}

[¼ - ¼ cos2_4x].cos2_{2x dx}

[¼ .cos2_{2x - ¼.cos}2_{2x .cos}2_{4x] dx}

{¼[½ + ½ cos 4x] - ¼ cos2 2x + ¼ sen2 4x.cos2_{2x} . dx}

{⅛ + ⅛ cos 4x - ¼ [½ +½ cos 2(2)x] + ¼ sen2_4x_[_½₊_{½ cos} 2(2)x]}.dx

{⅛ +⅛ cos 4x -¼ [½ + ½ cos 4x]+ ¼ sen2_4x_[_½₊_{½ cos 4x}_]}._dx

{⅛+⅛ cos 4x-⅛-⅛ cos 4x+⅛ sen2_4x₊_{⅛ sen}2_4x_._{cos 4x}_]}._dx

{⅛+⅛cos 4x-⅛-⅛ cos 4x+⅛ sen2_4x_{+ ⅛}_sen2_4x_._cos

4x]}.dx

[⅛ sen2_4x₊_{⅛ sen}2_4x_._{cos 4x}_]._dx

⅛ [½ -½ cos 2(4)x]dx + ⅛ (sen 4x)2.cos 4x.dx ⅛ [½ -½ cos 8x]dx + ⅛.¼ (sen 4x)2_{.cos 4x.(4)dx}

⅛ .½ dx - ⅛.½cos 8x dx + 1/32(sen 4x)2 .cos 4x.(4)dx

1/16dx - 1/16.⅛ cos 8x .(8)dx + 1/32 (sen 4x)2 .cos 4x.(4)dx

1/16x - 1/128 sen 8x + 1/32 (sen 4x)2+1 = 2+1 x - sen 8x + (sen 4x)3 = x + (sen 4x)3 - sen 8x = 16 128 32(3) 16 96 128 x + sen 3_{4x - sen 8x + c .} 16 96 128 10. (2 - sen θ)2_dθ = 9 θ + 4cos θ + sen 2 θ + c . 2 4 [4 - 2.2.sen θ + (sen θ)2_{] dθ} = [4 - 4sen θ + sen2 θ] dθ =

{8/2 + ½ - 4sen θ + ½ cos 2θ]}dθ = {9/2- 4sen θ + ½ cos 2θ]}dθ

9/2 dθ- 4sen θ .dθ + ½ cos 2θ .dθ =

9/2 dθ- 4sen θ .dθ + ½.½ cos 2θ .(2)dθ =

9/2 θ- 4(- cosθ) + ¼ (sen 2θ) = 9/2 θ+ 4cosθ + ¼ (sen 2θ)

θ + 4cos θ + sen 2 θ + c . 2 4

11. [sen2_{Ф + cos Ф]}2_d_Ф

[(sen2_Ф)2_{+ 2.(sen}2_{Ф).cos Ф + cos}2_Ф]2_d_Ф =

[(½ - ½ cos 2Ф)2_{+ 2.(sen}2_{Ф).cos Ф + (}_{½ + ½ cos 2}_{Ф)] d}_Ф

[(¼ - 2.½. ½ cos 2Ф + (½ cos 2Ф)2_{+ 2(sen}2_{Ф).cos Ф + ½ + ½ cos 2Ф]d} Ф =

[(¼ - ½ cos 2Ф + ¼ cos2_{2Ф + 2sen}2_{Ф.cos Ф + ½ + ½ cos 2Ф]d}

Ф =

[(¼ - ½ cos 2Ф + ¼ cos2_{2Ф + 2sen}2_{Ф.cos Ф + ½ + ½ cos 2Ф]d}

Ф =

[(¼ + ½ + ¼ cos2_{2Ф + 2sen}2_{Ф.cos Ф]d Ф}

[(¼ + ½ + ¼ (½ + ½ cos 2(2Ф) + 2sen2_{Ф.cos Ф]d Ф}

[(¼ + ½ + ⅛ + ⅛ cos 4Ф + 2sen2_{Ф.cos Ф]d Ф} =

[(2/8 + 4/8 + ⅛ + ⅛ cos 4Ф + 2sen2_{Ф.cos Ф]d Ф} =

[(7/8+ ⅛ cos 4Ф + 2sen2 Ф.cos Ф]d Ф =

7/8 dФ + ⅛.¼ cos 4Ф .(4Ф).d Ф + 2 (sen Ф)2.cos Ф.d Ф = 7/8 Ф + 1/32 sen 4Ф + 2 ( sen Ф ) 2+1 = 2+1 7 Ф + sen 4 Ф + 2(sen Ф ) 3 = 7 Ф + 2sen Ф3 + sen 4 Ф + c . 8 32 3 8 3 32

12. sen 2x cos 4x dx = cos 2x - cos 6x + c . 4 12

Por Trigonometría: sen 2x cos 4x = ½ sen[2+4]x + ½ sen[2-4]x

sen 2x cos 4x = ½ sen 6x + ½ sen[-2]x

{½ sen 6x + ½ sen[-2]x}dx = {½ sen 6x - ½ sen 2x}dx

½ sen 6x .dx - ½ sen 2x .dx =½.1/6 sen 6x.(6)dx - ½.½sen 2x .(2)dx

1/12 (- cos 6x) - ¼ (- cos 2x) = - cos 6x + cos 2x =

12 4

cos 2x - cos 6x + c .

4 12

13. sen 3x sen 2x dx = sen x - sen 5x + c . 2 10

Por Trigonometría: sen 3x sen 2x = -½ cos[3+2]x + ½ cos[3-2]x

sen 3x sen 2x = -½ cos 5x + ½ cos x

[-½ cos 5x + ½ cos x] dx = -½ cos 5x . dx + ½ cos x . dx =

-½.(1/5) cos 5x .(5) dx + ½ cos x .dx = -(1/10) sen 5x + ½ sen x =

2 10

14. cos 4x cos 3x dx

Por Trigonometría: cos 4x cos 3x = ½ cos[4+3]x + ½ cos[4-3]x

cos 4x cos 3x = ½ cos 7x + ½ cos x

(½ cos 7x + ½ cos x) dx = ½ cos 7x dx + ½ cos x dx

½.(1/7) cos 7x .(7)dx + ½ cos x dx

1/14(sen 7x) + ½ (sen x) = sen 7x + sen x = sen x + sen 7x + c

14 2 14 2 15. cos2_{ax dx}

[ ½ + ½ cos 2(ax) ]dx = [ ½ + ½ cos 2ax ]dx =

½ dx + ½ .1/2acos 2ax .(2a) ]dx =

x/2 + 1/4a sen 2ax = x/2 + sen 2ax /4a + c .

16. cos4_{ax dx} =

cos2_{ax . cos}2_{ax .dx}

= [ ½ + ½ cos 2ax] [½ + ½ cos 2ax] dx =

[½ + ½ cos 2ax]2_dx

= [¼ + 2.½.½ cos 2ax + ¼ cos2 2ax]dx =

{¼ + ½ cos 2ax + ¼ [½ + ½ cos 2(2ax)]}dx =

{¼ + ½ cos 2ax + ⅛ + ⅛ cos 4ax}dx .Haciendo operaciones:

{⅜ + ½ cos 2ax + ⅛ cos 4ax}dx

⅜ dx + ½ .1/2acos 2ax .(2a) dx + ⅛.

¼

a cos 4ax .(4a)}dx 3x/8 + 1/4a sen 2ax + 1/32a sen4ax + c .

17. sen2_{ax . cos}2_{ax .dx} =

[ ½ - ½ cos 2(ax)] [½ + ½ cos 2(ax)] dx =

[(½)2_{- (½ cos}_2ax)2_{] dx}

= [¼ - ¼ [½ + ½ cos 2(2ax)2] dx =

[¼ - ⅛ - ⅛ cos 4ax] dx = [2/8 - ⅛ - ⅛ cos 4ax] dx =

[⅛ - ⅛ cos 4ax] dx = ⅛dx - ⅛.¼a cos 4ax.(4a)] dx =

x/8 - 1/32a .sen 4ax = x/8 - sen 4ax/32a + c .

18. sen4__{/2 cos}2__{/2 .d}_ =

sen2__/2_.cos2__/2_.sen2__/2_d_

= (sen /2 .cos /2)2 .sen2/2 d=

Por Trigonometría:sen 2x = 2senx.cosx ; sen /2 .cos /2 = sen(2./2) sen /2 .cos /2= ½ sen .

(sen /2 .cos /2)2 .sen2/2 d=

(½sen. sen 2__/2)_d_

= {(½sen  [½ - ½ cos (2./2)]} d=

[½sen  ( ½ - ½ cos )] d= [¼ sen - ¼ sen .cos ] d=

¼ sen d - ¼ (sen )1.cos _ d .

v = sen  El diferencial esta completo, se procede a integrar.

dv = cos  d Se usa: vn dv = v n+1 + c . n = 1 n+1 ¼ (- cos ) - ¼ .sen 2_ = - cos  - sen 2  + c . 2 4 8 19.  csc ax 4_{. dx} = cot ax

1 .

Por Trigonometría: csc ax = sen ax = 1 = sec ax

cot ax cos ax cos ax . sen ax

 csc ax 4_.dx

= (sec ax)4.dx = sec4 ax.dx =sec2 ax. sec2 ax dx = cot ax

(1 + tg2_{ax). sec}2_{ax . dx}

= (sec2 ax + tg2 ax . sec2 ax). dx =

sec2_{ax . dx +}__tg2_{ax . sec}2_{ax). dx} =

sec2_{ax . dx +}__{(tg ax)}2_{. sec}2_{ax. dx} =

v = ax 1 ra integral : Falta (a) para completar el diferencial.

dv = a dx Se aplica: sec2 v = tg v + c .

v = tg ax 2 da integral : Falta (a) para completar el diferencial.

dv = a.sec2 ax dx Se aplica: vn dv = v n+1 + c .

n+1 . 1 .sec2_{ax .(a) dx + . 1 .}__{(tg ax)}2_{.(a) sec}2_{ax. dx}

= a a tg ax + (tg ax)2+1 = tg ax + (tg ax)2+1 + c . a (2+1)a a 3a 20. sen2_{x . cos}6_{x . dx .}

sen2_{x . cos}2_{x . cos}2_{x . cos}2_{x . dx .}

(sen x . cos x)2_{.(cos x . cos x)}2_{. dx .}

Por trigonometría: sen x.cos x = sen 2x ;

cos2_x

= cos 2x + 1 = ½ cos 2x + ½ .Sustituyendo en la integral .

(sen x . cos x)2_{.(cos x . cos x)}2_{. dx .}

( ½ sen 2x)2_{.( ½ cos 2x + ½)}2_{. dx .}

(½ sen 2x)2 .( ½ cos 2x + ½)2 . dx .Haciendo operaciones. (1/4 sen2 2x) .[1/4 cos2 2x + 2. ½ cos 2x . ½ + ¼] . dx .

(1/4 sen2 2x) .[1/4 cos2 2x + ½ cos 2x + ¼] . dx .

[1/16 sen2_{2x . cos}2_{2x +}_1/8_sen2_{2x . cos 2x +}_1/16_sen2_{2x].dx .}

1/16 sen2_{2x . cos}2_{2x + 1/8 . ½}__{(sen 2x)}2_{.cos 2x.(2)+ 1/16}__sen2_{2x.dx .}

1/16sen2_2x.cos2_{2x +1/16}_{(sen 2x)}2_{.cos 2x.(2)+ 1/16}__{[1/2 - ½ cos 2(2x) ] .dx .} 1/16(sen 2x .cos 2x)2 _{dx + ( sen 2x)}2+1_{+ 1/16.1/2}_dx_{- 1/16}_._{½ .1/4}_{cos 4x .(4)dx .}

16(2+1)

1/16[1/2sen 2(2x)]2 _{dx + ( sen}3_{2x) + 1/32}_x_{- 1/128}_{cos 4x .(4)dx .}

16(3)

1/16[1/4sen2_4x)]_{dx + ( sen}3_{2x) + 1/32}_x_{- 1/128 sen 4x .}

1/16 . 1/4 [sen2_4x]._{dx + ( sen}3_{2x) + 1/32}_x_{- 1/128 sen 4x .}

1/16 . 1/4 [1/2 – ½ cos 2(4x)]dx + ( sen 3_{2x) + 1/32}_x_{- 1/128 sen 4x .}

1/64 [1/2 – ½ cos 8x]dx + ( sen 3_{2x) + 1/32}_x_{- 1/128 sen 4x .}

1/64 . 1/2 dx – 1/64 . ½ cos 8x dx + ( sen _{2x) + 1/32}3 _x_{- 1/128 sen 4x .}

1/128 x – 1/128 . 1/8 cos 8x . (8)dx + ( sen 3_{2x) + 1/32}_x_{- 1/128 sen 4x .}

5/128 x - 1/1024 sen 8x + (sen3_{2x) - 1/128 sen 4x + c .}

21. (1 + cos x)3_{. dx .}

(13_{+ 3.1}2_{.cos x + 3.1.cos}2_{x + cos}3_{x) . dx .}

(1 + 3cos x + 3cos2_{x + cos}3_{x) . dx .}

[1 + 3cos x + 3(½ + ½ cos 2x) + cos2_{x .cos x] . dx .}

[1 + 3cos x + 3(½ + ½ cos 2x) + (1 - sen2 x) .cos x] . dx .

[2/2 + 3cos x + 3/2 + 3/2 cos 2x + cos x - sen2_{x .cos x] . dx .}

[5/2 + 4cos x + 3/2 cos 2x - sen2 x .cos x] . dx .

5/2 dx + 4 cos x dx + 3/2 . ½ cos 2x .(2) dx - (sen x)2_{.cos x . dx .}

2+1 5x + 4 sen x + 3sen 2x - (sen x)3_{+ c .} 2 4 3

22. (√sen 2θ - cos 2θ)2_dθ

[√sen 2θ )2_{- 2}_{(√sen 2}_{θ ) . cos}₂_{θ + cos}2₂_{θ ] dθ}

[sen 2θ - 2(sen 2θ )1/2_{. cos}₂_{θ + cos}2₂_{θ ] dθ}

{sen 2θ - 2(sen 2θ )1/2_{. cos}₂_{θ + [}_1/2₊_½_{cos 2(}₂_{θ)]} dθ}

{sen 2θ - 2(sen 2θ )1/2_{. cos}₂_{θ +}_1/2₊_½_{cos 4θ} dθ}

sen 2θ . dθ - 2.½ (sen 2θ )1/2_{.cos 2θ.(2) + 1/2}__{dθ + ½ .1/4}__{cos 4θ.(4) dθ}

½ sen 2θ.(2). dθ - (sen 2 θ ) 1/2+1 _{+ θ/}₂₊_1/8_{sen 4θ dθ}

½+1

½ (- cos 2θ) - (sen 2 θ ) 1/2+1 _{+ θ/}₂₊_1/8_{sen 4θ dθ}

½+1

- ½ (cos 2θ) - (sen 2 θ ) 3/2 _{+ θ/}₂₊_1/8_{sen 4θ dθ}

3/2

- ½ (cos 2θ) - 2(sen 2 θ ) 3/2 _{+ θ/}₂₊_1/8_{sen 4θ + c .} 3

Ordenando:

θ/2 + 1/8 sen 4θ- 2(sen 2 θ ) 3/2 _-_½_{(cos 2}_{θ) + c .}

3 23. (√cos θ - 2sen θ)2_dθ

[(√cos θ )2_{- 2}_(√cos_{θ .2sen θ + (2 sen θ)}2_{] dθ}

[(cos θ ) - 2.2 (cos θ)1/2_{.sen θ + (4 sen}2_{θ)] dθ}

[cos θ - 4(cos θ)1/2_{.sen θ + 4(}_1/2_-_1/2_{cos 2θ] dθ}

[(cos θ ) - 4(cos θ)1/2_{.sen θ + 4/2 -}_4/2_{cos 2θ] dθ}

[cos θ - 4(cos θ)1/2_{.sen θ + 2 -}₂_{cos 2θ] dθ}

(cos θ ) dθ - 4(cos θ)1/2_{.sen θ dθ + 2}__{dθ -}_2.½__{cos 2θ.(2) .dθ}

sen θ - 4 ( cos θ ) 1/2+1 _{+ 2θ -}_sen_2θ ½+1

sen θ - 4 ( cos θ ) 3/2 _{+ 2θ -}_sen_{2θ + c .} 3/2

sen θ - 8 ( cos θ ) 3/2 _{+ 2θ -}_sen_{2θ + c .} 3

24. (sen 2x - sen 3x)2_dx

(sen2_{2x - 2 sen 2x .sen 3x + sen}2_3x)dx

{(½ - ½ cos 2(2x) - 2[- ½ cos(2+3)x + ½ cos(2-3)x] + [½ - ½ cos 2(3x)]dx

{(½ - ½ cos 4x) - 2[- ½ cos 5x + ½ cos (-x)] + [½ - ½ cos 6x]}dx

{½ - ½ cos 4x + cos 5x - cos (-x) + ½ - ½ cos 6x}dx Por Trigonometría: cos (-x) = cos (x) .

{½ + ½ - ½ cos 4x + cos 5x + cos x - ½ cos 6x}dx

{1 - ½ cos 4x + cos 5x + cos x - ½ cos 6x}dx

dx - ½.1/4 cos 4x .(4) dx + 1/5 cos 5x .(5)dx + cos x.dx - ½ .1/6 cos 6x .(6)dx

x dx -1/8 cos 4x .(4) dx + 1/5 cos 5x .(5)dx + cos x.dx - 1/12 cos 6x .(6)dx

x -1/8 sen 4x + 1/5 sen 5x + sen x - 1/12 sen 6x . x - sen 4x + sen 5x + sen x - sen 6x + c .

8 5 12 25. (sen x + cos 2x)2_dx

(sen2_{x + 2 sen x .cos 2x + cos}2_{2x) dx}

Por Trigonometría: cos 2x = cos2x – sen2x ; sen2x = ½-½cos 2x ; cos2x = ½+½ cos 2x.

[(½ - ½ cos 2x) + 2 sen x (cos2_{x - sen}2_{x) + ½ + ½ cos 2(2x)] dx}

[½ - ½ cos 2x + 2 cos2 _{x .sen x - 2sen}2 _{x. senx + ½ + ½ cos 4x] dx}

[½ + ½ - ½ cos 2x + 2 cos2 _{x .sen x - 2sen}2 _{x. senx + ½ cos 4x] dx}

[1 - ½ cos 2x + 2 cos2 _{x .sen x - 2(1 - cos}2_{x).senx + ½ cos 4x] dx}

[1 - ½ cos 2x + 2 cos2 _{x .sen x - 2 sen x + 2cos}2 _{x.senx + ½ cos 4x] dx}

[1 - ½ cos 2x + 4 cos2 _{x .sen x - 2 sen x + ½ cos 4x] dx .}

dx - ½ . ½ cos 2x .(2)dx + (-)4(cosx)2_{.(-)sen x dx}_-₂_{sen x .dx}_{+ ½.}_1/4

cos 4x.(4) dx

dx -1/4cos 2x .(2)dx - 4(cosx)2_{.(-)sen x dx}-₂_{sen x .dx + 1/8}_{cos 4x.(4) dx} x-sen 2x -4(cos x)3-2 (- cos x)+ sen 4x + c .

4 3 8 x-sen 2x -4(cos x)3+ 2 cos x+ sen 4x + c .

4 3 8

26. (cos x + 2cos 2x)2_dx

[(cos x)2_{+ 2(cos x).(2cos 2x) + (2 cos 2x)}2_dx

(cos2_{x + 4 cos x .cos 2x + 4cos}2_{2x) dx}

(½ + ½ cos 2x + 4 cos x .(cos2_{x - sen}2_{x) + 4(½ + ½ cos 2(2x) dx}

½ + 2 + ½ cos 2x + 4 cos2_{x. cos x - 4sen}2_{x.cos x + 2 cos 4x] dx}

5/2 + ½ cos 2x + 4 cos2 x. cos x - 4sen2 x.cos x + 2 cos 4x] dx

[5/2 + ½ cos 2x + 4(1 - sen2 x) . cos x - 4sen2 x.cos x + 2 cos 4x] dx

[5/2 + ½ cos 2x + 4cos x - 4sen2_{x .cos x - 4sen}2_{x.cos x + 2 cos 4x] dx}

[5/2 + ½ cos 2x + 4cos x - 8sen2_{x .cos x + 2 cos 4x] dx}

5/2dx + ½.½cos 2x.(2)dx + 4(-)cos x.(-)dx - 8(sen x)2_{.cos x.dx}+ 2.1/4cos 4x.(4)dx

5/2dx + 1/4 cos 2x.(2)dx - 4 cos x.(-)dx - 8(sen x)2_{.cos x.dx + ½}__{cos 4x.(4)dx}

5x + sen 2x - 4sen x – 8(sen x)2+1_{+ sen 4x + c .}

2 4 2+1 2

5x + sen 2x - 4sen x - 8(sen x)3_{+ sen 4x + c .} 2 4 3 2

In document Propensity score matching methods for the analysis of recurrent events (Page 33-40)