INTEGRALES INMEDIATAS (DIRECTAS)

INTEGRALES INMEDIATAS (DIRECTAS)

1.- ∫cosx sen^3 xdx

(sen^4 x)/4+c

2.-∫2x eˣ²dx

eˣ²+c

3.-∫x dx/(〖(x〗^2+3)⁵) dx

(-1)/(8〖(x〗^2+3)⁴)+c

4.-∫1/x ln^3 xdx

(ln^4⎸x⎸)/4+ c

5.-∫dx/(x-1)

ln⎸x-1⎸+c

6.-∫(x+√x)/x^2 dx

ln⎸x⎸-2/√x+ c

7.- ∫ 3/(1+x²) dx

3 arc tg x+c

8.-∫dx/(x-4)^2

(-1)/((x-4))+ c

9.-∫dx/(x-4)

ln⎸x-4⎸+c

10.- ∫eˣ^(-4) dx

eˣ-^4+ c

11.- ∫e^(-2) ˣ^(+9) dx

-1/2 e ⁻²ˣ^(+9)+ c

12.- ∫ dx/(x+4)

1/2 arc tg (X/2)+ c

13.-∫4/(3+x^2 ) dx

(4√3)/3 arc tg (x/√3)+ c

14.-∫5/(4x^2+1) dx

5/2 arc tg (2x)+ c

15.-∫2/(1+9x^2 ) dx

2/3 arc tg (3x)+ c

16.-∫dx/(√4-x^2 ) dx

arc sen (x/2)+ c

17.-∫ eˣ/(√1-e^2 ˣ) dx

arc sen (eˣ)+ c

INTEGRALES POR PARTES

∫u.v΄dx=u.v- ∫u΄.vdx

1.- ∫x cos⁡〖x dx〗=x sen x- ∫sen x dx

=x senx+cosx+c

U: x derivar du: 1

V΄: cos x integrar V: sen x

2.- ∫senx [ln(cosx) ]dx= -cosx ln⁡〖(cosx)- ∫senx dx〗

=-cosx ln⁡〖(cosx)+cos⁡〖x+c〗 〗

U: ln(cosx) U΄: senx/cosx

V΄: senx dv: - cosx

3.-∫x-tan⁡〖x dx= -x ln⁡〖cosx+ ∫ln⁡〖cosx dx〗 〗 〗

=-x•ln⁡〖cosx+tan⁡〖x+c〗 〗

U: x du:1

V΄: tanx dv: -ln cos x

4.-∫x ln⁡〖2x=ln⁡〖2x •1/2 x²-∫▒〖1/2 x²;1/x dx〗〗 〗

= 1/2 x² ln⁡〖2x-1/2∫x 1/x dx〗

= 1/2 x² ln⁡〖2x-1/4 x²+c〗

U: ln 2x du: 1/x

V΄: x V:1/2 x²

5.- ∫x^2 ln⁡〖x dx=lnx •2x-∫1/x〗• 2x dx

=2x ln⁡〖x-2 ∫x 1/x dx〗

=2x ln⁡〖x-2x+c〗

U: ln x du:1/x

V: x² V: 2x

6.-∫x^4 lnx dx= ∫lnx x^4 dx

=lnx 1/5 x^5-∫1/x•1/5 x⁵dx

=lnx•1/5 x^5-1/5∫x⁴dx

=lnx•1/5 x^5-1/5 (1/5 x^5 )

=lnx 1/5 x^5-1/25 x^5+ c

U: lnx du:1/x

V΄: x⁴ dv:1/5 x⁵

7.- ∫lnx dx=lnx•x-∫1/x•x dx

=x lnx-∫1•dx

=x ln⁡〖x-x+c〗

U: lnx du:1/x

V΄: 1 dv: x

8.-∫x sec^2 dx=x•tanx- ∫1•tanxdx

=x•tanx+ln⁡〖(cosx)+ c〗

U: x du: 1

V΄: sec² dv: tanx

9.- ∫arc cotg x dx= ∫arc cotg x•1 dx

=arc cotg x•x- ∫- 1/(1+x^2 )•xdx

=x arc cotg x+∫x/(1+x^2 ) dx

=x arc cotg x+1/2 ln⁡(1+x^2 )+c

U: arc cot x du: 1/(1+x^2 )

V΄:1 dv: x

10.-∫(2x-5)senx dx=(2x-5)(-cosx)+2∫cosx dx

=-(2x-5)cosx+2 senx+c

U: 2x-5 du:2

V΄: senx dv: -cosx

11.-x e^5 ˣ dx=x•(e^5 ˣ)/5-(∫e^5 ˣ)/5 dx=(xe^5 ˣ)/5-1/5 ((e^5 ˣ)/5)

= (xe^5 ˣ)/5-(e^5 ˣ)/25=((5xe^5 ˣ-e^5 ˣ)/25)

= (e^5 ˣ(5x-1))/25+c

U: x du: 1

V: e⁵ˣ dv: e⁵ˣ/5

12.-sen^-

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