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Question Number 55520 by Rdk96 last updated on 26/Feb/19
How can solve ∫(√)tan(x)dx ?
Howcansolvetan(x)dx?
Commented by maxmathsup by imad last updated on 26/Feb/19
let A =∫ (√(tanx))dx  changement (√(tanx))=t give tanx =t^2  ⇒x =arctan(t^2 ) ⇒  A =∫ t ((2t)/(1+t^4 )) dt =2 ∫  (t^2 /(t^4 +1))dt  let decompose F(t)=(t^2 /(t^4  +1)) ⇒  F(t)=(t^2 /((t^2  −(√2)t +1)(t^2  +(√2)t +1))) =((at +b)/(t^2 −(√2)t +1)) +((ct +d)/(t^2  +(√2)t +1))  F(−t)=F(t) ⇒((−at +b)/(t^2 +(√2)t +1))+((−ct +d)/(t^2 −(√2)t +1)) =F(t) ⇒c=−a and d=b ⇒  F(t)=((at +b)/(t^2 −(√2)t+1)) +((−at +b)/(t^2  +(√2)t +1))  F(0) =0 =2b ⇒b=0 ⇒F(t)=((at)/(t^2 −(√2)t +1)) −((at)/(t^2 +(√2)t +1))  F(1) =(1/((2−(√2))(2+(√2)))) = (a/(2−(√2))) −(a/(2+(√2))) ⇒(1/2) =(((2+(√2))a−(2−(√2))a)/2) ⇒  2(√2)a =1 ⇒a =(1/(2(√2))) ⇒F(t)=(1/(2(√2))){ (t/(t^2 −(√2)t +1)) −(t/(t^2  +(√2)t +1))} ⇒  A =(1/( (√2))){ ∫   ((tdt)/(t^2 −(√2)t +1)) −∫   ((tdt)/(t^2 +(√2)t +1))} +c  but   ∫   ((tdt)/(t^2 −(√2)t +1)) =(1/2) ∫  ((2t−(√2)+(√2))/(t^2 −(√2)t +1)) dt =(1/2)ln(t^2 −(√2)t +1)+(1/( (√2)))∫  (dt/(t^2 −(√2)t +1))  ∫  (dt/(t^2 −(√2)t +1)) =∫  (dt/((t−((√2)/2))^2  +(1/2))) =_(t−(1/( (√2)))=(u/( (√2))))     ∫     (du/( (√2)(1/2)(1+u^2 )))  =(√2)arctan(t(√2)−1) ⇒∫  ((tdt)/(t^2 −(√2)t +1)) =(1/2)ln(t^2 −(√2)t +1) +arctan(t(√2)−1)  also we get  ∫  ((tdt)/(t^2 +(√2)t +1)) =(1/2)ln(t^2  +(√2)t +1) +arctan(t(√2)+1) ⇒  (√2)A =(1/2)ln(t^2 −(√2)t +1)+arctan(t(√2)−1)−(1/2)ln(t^2  +(√2)t +1)−arctan(t(√2)+1) +c  =ln(√((t^2 −(√2)t+1)/(t^2  +(√2)t +1)))  +arctan(t(√2)−1)−arctan(t(√2)+1) +c ⇒  A =(1/( (√2))){ ln((√((tanx−(√(2tanx))+1)/(tanx+(√(2tanx))+1)))) +arctan((√(2tanx))−1)−arctan((√(2tanx)) +1)} +c
letA=tanxdxchangementtanx=tgivetanx=t2x=arctan(t2)A=t2t1+t4dt=2t2t4+1dtletdecomposeF(t)=t2t4+1F(t)=t2(t22t+1)(t2+2t+1)=at+bt22t+1+ct+dt2+2t+1F(t)=F(t)at+bt2+2t+1+ct+dt22t+1=F(t)c=aandd=bF(t)=at+bt22t+1+at+bt2+2t+1F(0)=0=2bb=0F(t)=att22t+1att2+2t+1F(1)=1(22)(2+2)=a22a2+212=(2+2)a(22)a222a=1a=122F(t)=122{tt22t+1tt2+2t+1}A=12{tdtt22t+1tdtt2+2t+1}+cbuttdtt22t+1=122t2+2t22t+1dt=12ln(t22t+1)+12dtt22t+1dtt22t+1=dt(t22)2+12=t12=u2du212(1+u2)=2arctan(t21)tdtt22t+1=12ln(t22t+1)+arctan(t21)alsowegettdtt2+2t+1=12ln(t2+2t+1)+arctan(t2+1)2A=12ln(t22t+1)+arctan(t21)12ln(t2+2t+1)arctan(t2+1)+c=lnt22t+1t2+2t+1+arctan(t21)arctan(t2+1)+cA=12{ln(tanx2tanx+1tanx+2tanx+1)+arctan(2tanx1)arctan(2tanx+1)}+c
Commented by maxmathsup by imad last updated on 26/Feb/19
∫  ((tdt)/(t^2 +(√2)t +1)) =_(t=−u)    ∫  (((−u)(−du))/(u^2 −(√2)u +1)) =∫ ((udu)/(u^2 −(√2)u +1))  =(1/2)ln(u^2 −(√2)u +1) +arctan(u(√2)−1)  =(1/2)ln(t^2 +(√2)t +1)−arctan(t(√2)+1) ⇒  (√2)A =(1/2)ln(t^2 −(√2)t +1)+arctan(t(√2)−1)−(1/2)ln(t^2  +(√2)t +1) +arctan(t(√2)+1)  A =(1/( (√2))){ ln((√((tanx−2(√(tanx))+1)/(tanx +(√(2tanx))+1)))) +arctan(t(√2)−1) +arctan(t(√2)+1)} +c .
tdtt2+2t+1=t=u(u)(du)u22u+1=uduu22u+1=12ln(u22u+1)+arctan(u21)=12ln(t2+2t+1)arctan(t2+1)2A=12ln(t22t+1)+arctan(t21)12ln(t2+2t+1)+arctan(t2+1)A=12{ln(tanx2tanx+1tanx+2tanx+1)+arctan(t21)+arctan(t2+1)}+c.
Answered by tanmay.chaudhury50@gmail.com last updated on 26/Feb/19
t^2 =tanx   2tdt=sec^2 xdx  ((2tdt)/(1+t^4 ))=dx  ∫t×((2tdt)/(1+t^4 ))  ∫((2t^2 )/(1+t^4 ))dt  ∫((2dt)/(t^2 +(1/t^2 )))  ∫((1−(1/t^2 )+1+(1/t^2 ))/(t^2 +(1/t^2 )))dt←main step  ∫((1−(1/t^2 ))/((t+(1/t))^2 −2))dt+∫((1+(1/t^2 ))/((t−(1/t))^2 +2))dt  ∫((d(t+(1/t)))/((t+(1/t))^2 −2))+∫((d(t−(1/t)))/((t−(1/t))^2 +2))  now use formula ∫(dx/(x^2 −a^2 ))=(1/(2a))ln(((x−a)/(x+a)))+c  and∫(dx/(x^2 +a^2 ))=(1/a)tan^(−1) ((x/a))+c_1   (1/(2(√2)))ln(((t+(1/t)−(√2))/(t+(1/t)+(√2))))+(1/( (√2)))tan^(−1) (((t−(1/t))/( (√2))))+c  (1/(2(√2)))ln((((√(tanx)) +(1/( (√(tanx)) ))−(√2))/( (√(tanx)) +(1/( (√(tanx)) ))+(√2))))+(1/( (√2)))tan^(−1) ((((√(tanx)) −(1/( (√(tanx)) )))/( (√2))))+c  pls check...
t2=tanx2tdt=sec2xdx2tdt1+t4=dxt×2tdt1+t42t21+t4dt2dtt2+1t211t2+1+1t2t2+1t2dtmainstep11t2(t+1t)22dt+1+1t2(t1t)2+2dtd(t+1t)(t+1t)22+d(t1t)(t1t)2+2nowuseformuladxx2a2=12aln(xax+a)+canddxx2+a2=1atan1(xa)+c1122ln(t+1t2t+1t+2)+12tan1(t1t2)+c122ln(tanx+1tanx2tanx+1tanx+2)+12tan1(tanx1tanx2)+cplscheck
Commented by Rdk96 last updated on 26/Feb/19
it is possible to multiply the differential   with the equation??
itispossibletomultiplythedifferentialwiththeequation??
Commented by tanmay.chaudhury50@gmail.com last updated on 26/Feb/19
yes...  ∫(x/(1+x^2 ))dx→(1/2)∫((d(1+x^2 ))/(1+x^2 ))→(1/2)ln(1+x^2 )+c
yesx1+x2dx12d(1+x2)1+x212ln(1+x2)+c
Commented by Rdk96 last updated on 26/Feb/19
correct.
correct.
Answered by tanmay.chaudhury50@gmail.com last updated on 26/Feb/19
or method...  p=∫(√(tanx)) +(√(cotx)) dx  q=(√(tanx)) −(√(cotx)) dx  ∫(√(tanx)) dx=((p+q)/2)  p=∫(√((sinx)/(cosx))) +(√((cosx)/(sinx))) dx  p=∫((sinx+cosx)/( (√(sinxcosx))))dx  p=(√2)∫((d(sinx−cosx))/( (√(1−1+2sinxcosx))))dx  =(√2) ∫((d(sinx−cosx))/( (√(1−(sinx−cosx)^2 )) ))  =(√2) sin^(−1) (sinx−cosx) +c_1   q=∫((√((sinx)/(cosx))) −(√((cosx)/(sinx))) ) dx  q=∫((sinx−cosx)/( (√(sinxcosx))))dx→−∫((cosx−sinx)/( (√(sinxcosx))))dx  q=(−(√2) )∫((d(sinx+cosx))/( (√(1+2sinxcosx−1))))  =(−(√2) )∫((d(sinx+cosx))/( (√((sinx+cosx)^2 −1))))  =(−(√2) )ln{(sinx+cosx)+(√((sinx+cosx)^2 −1)) }    so ∫(√(tanx)) dx  ((√2)/2)[sin^(−1) (sinx−cosx)−ln{(sinx+cosx)+(√((sinx+cosx)^2 −1)) }]+c  pls check...
ormethodp=tanx+cotxdxq=tanxcotxdxtanxdx=p+q2p=sinxcosx+cosxsinxdxp=sinx+cosxsinxcosxdxp=2d(sinxcosx)11+2sinxcosxdx=2d(sinxcosx)1(sinxcosx)2=2sin1(sinxcosx)+c1q=(sinxcosxcosxsinx)dxq=sinxcosxsinxcosxdxcosxsinxsinxcosxdxq=(2)d(sinx+cosx)1+2sinxcosx1=(2)d(sinx+cosx)(sinx+cosx)21=(2)ln{(sinx+cosx)+(sinx+cosx)21}sotanxdx22[sin1(sinxcosx)ln{(sinx+cosx)+(sinx+cosx)21}]+cplscheck
Answered by MJS last updated on 26/Feb/19
∫(√(tan x))dx=       [t=(√(tan x)) → dx=2cos^2  x (√(tan x))dt=((2tdt)/(t^4 +1))]  =2∫(t^2 /(t^4 +1))dt=2∫(t^2 /((t^2 −(√2)t+1)(t^2 +(√2)t+1)))dt=  =2∫((((√2)t)/(4(t^2 −(√2)t+1)))−(((√2)t)/(4(t^2 +(√2)t+1))))dt=  =(1/2)∫((((√2)t−1)/(t^2 −(√2)t+1))+(1/(t^2 −(√2)t+1))−(((√2)t+1)/(t^2 +(√2)t+1))+(1/(t^2 +(√2)t+1)))dt=  =((√2)/4)∫(((2t−(√2))/(t^2 −(√2)t+1))−((2t+(√2))/(t^2 +(√2)t+1)))dt+(1/2)∫((1/(t^2 −(√2)t+1))+(1/(t^2 +(√2)t+1)))dt=  =((√2)/4)(ln (t^2 −(√2)t+1) −ln (t^2 +(√2)t+1))+((√2)/2)(arctan ((√2)t−1) +arctan ((√2)t+1))=  =((√2)/4)(ln ∣((tan x −(√(2tan x))+1)/(tan x +(√(2tan x))+1))∣ +2(arctan ((√(2tan x))−1) +arctan ((√(2tan x))+1)))+C
tanxdx=[t=tanxdx=2cos2xtanxdt=2tdtt4+1]=2t2t4+1dt=2t2(t22t+1)(t2+2t+1)dt==2(2t4(t22t+1)2t4(t2+2t+1))dt==12(2t1t22t+1+1t22t+12t+1t2+2t+1+1t2+2t+1)dt==24(2t2t22t+12t+2t2+2t+1)dt+12(1t22t+1+1t2+2t+1)dt==24(ln(t22t+1)ln(t2+2t+1))+22(arctan(2t1)+arctan(2t+1))==24(lntanx2tanx+1tanx+2tanx+1+2(arctan(2tanx1)+arctan(2tanx+1)))+C
Commented by behi83417@gmail.com last updated on 26/Feb/19
arctg((√(2tgx))−1)+arctg((√(2tgx))+1)=  arctg((2(√(2tgx)))/(2(1−tgx)))=arctg(((√(2tgx))/(1−tgx)))=  arccos∣cosx−sinx∣  ln((tgx−(√(2tgx))+1)/(tgx+(√(2tgx))+1))=ln(((tgx+1−(√(2tgx)))^2 )/(tg^2 x+1))=  =2ln[cosx.(tgx+1−(√(2tgx))]=  =2ln[sinx+cosx−(√(sin2x))]  ⇒I=((√2)/2).ln∣cosx+sinx−(√(sin2x))∣+2cos^(−1) ∣cosx−sinx∣+C
arctg(2tgx1)+arctg(2tgx+1)=arctg22tgx2(1tgx)=arctg(2tgx1tgx)=arccoscosxsinxlntgx2tgx+1tgx+2tgx+1=ln(tgx+12tgx)2tg2x+1==2ln[cosx.(tgx+12tgx]==2ln[sinx+cosxsin2x]I=22.lncosx+sinxsin2x+2cos1cosxsinx+C
Commented by MJS last updated on 26/Feb/19
thank you!
thankyou!

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