Menu Close

1-x-2-x-1-2-x-2-x-2-x-1-dx-




Question Number 125194 by bemath last updated on 09/Dec/20
 ∫ (((1−(√(x^2 +x+1)))^2 )/(x^2  (√(x^2 +x+1)))) dx ?
(1x2+x+1)2x2x2+x+1dx?
Answered by liberty last updated on 09/Dec/20
 by second Euler substitution    we set (√(x^2 +x+1)) = xt + 1 ; then    x^2 +x+1 = x^2 t^2  + 2xt + 1 ; x=((2t−1)/(1−t^2 ))   dx = ((2t^2 −2t+2)/((1−t^2 )^2 )) dt ;   I= ∫(((1−(√(x^2 +x+1)) )^2 )/(x^2  (√(x^2 +x+1)))) dx = 2∫ (t^2 /(1−t^2 )) dt   I = −2t + ln ∣ ((1+t)/(1−t)) ∣ + c   I = −((2((√(x^2 +x+1)) −1))/x) + ln ∣ ((x+(√(x^2 +x+1))−1)/(x−(√(x^2 +x+1)) +1)) ∣ + c  = −((2((√(x^2 +x+1)) −1))/x) + ln ∣2x+2(√(x^2 +x+1)) +1 ∣ + c
bysecondEulersubstitutionwesetx2+x+1=xt+1;thenx2+x+1=x2t2+2xt+1;x=2t11t2dx=2t22t+2(1t2)2dt;I=(1x2+x+1)2x2x2+x+1dx=2t21t2dtI=2t+ln1+t1t+cI=2(x2+x+11)x+lnx+x2+x+11xx2+x+1+1+c=2(x2+x+11)x+ln2x+2x2+x+1+1+c

Leave a Reply

Your email address will not be published. Required fields are marked *