Question Number 70253 by oyemi kemewari last updated on 02/Oct/19

Commented bymathmax by abdo last updated on 02/Oct/19

let I =∫ u^2 (√(u^2 −2))du changement u=(√2)ch(x) give I =∫ 2ch^2 (x)(√2)sh(x)(√2)sh(x)dx =4 ∫ ch^2 x sh^2 x dx =4 ∫ (ch(x)sh(x))^2 dx =4 ∫((1/2)sh(2x))^2 dx =∫ sh^2 (2x)dx =∫((ch(4x)−1)/2)dx =−(x/2) +(1/8)sh(4x)dx +C =−(x/2) +(1/8){((e^(4x) −e^(−4x) )/2)} +C =−(x/2) +(1/(16))(e^(4x) −e^(−4x) ) +C but x=argch((u/(√2))) =ln((u/(√2))+(√((u^2 /2)−1))) ⇒ I =−(1/2)ln((u/(√2))+(√((u^2 /2)−1)))+(1/(16)){((u/(√2))+(√((u^2 /2)−1)))^4 −((u/(√2))+(√((u^2 /2)−1)))^(−4) }+C =

Commented byoyemi kemewari last updated on 02/Oct/19

thanks you slr

Commented bymathmax by abdo last updated on 03/Oct/19

you are welcome.