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Question Number 44781 by arvinddayama01@gmail.com last updated on 04/Oct/18
prove that:− ∫(1/(t(√(1−t^2 ))))dt = ln(1−(√(1−t^2 )))+C
provethat:−∫1t1−t2dt=ln(1−1−t2)+C
Answered by tanmay.chaudhury50@gmail.com last updated on 04/Oct/18
∫((tdt)/(t^2 (√(1−t^2 )) )) k^2 =1−t^2 2kdk=−2tdt ∫((−kdk)/((1−k^2 )k)) ∫(dk/(k^2 −1)) (1/2)∫(((k+1)−(k−1))/((k+1)(k−1)))dk (1/2)∫(dk/(k−1))−(1/2)∫(dk/(k+1)) (1/2){ln(k−1)−ln(k+1)}+c (1/2){ln(((k−1)/(k+1)))}+c (1/2){ln((((√(1−t^2 )) −1)/( (√(1−t^2 )) +1)))}+c←correct answer recheck of answer given RHS (d/dt){ln(1−(√(1−t^2 )) )} (1/(1−(√(1−t^2 )) ))×((0−(1/(2(√(1−t^2 )) ))×−2t)/1) (1/(1−(√(1−t^2 )) ))×(t/( (√(1−t^2 )))) now recheck result obtained (d/dt)[(1/2){ln((√(1−t^2 )) −1)−ln((√(1−t^2 )) +1)}] (1/2)[{(1/( (√(1−t^2 )) −1))×(1/(2(√(1−t^2 )) ))×−2t}−{(1/( (√(1−t^2 )) +1))×((−2t)/(2(√(1−t^2 )) ))}] (1/2)[((−t)/( (√(1−t^2 )) .((√(1−t^2 )) −1)))+(t/( (√(1−t^2 )) .((√(1−t^2 )) +1)))] (t/(2(√(1−t^2 )) )){((−1)/( (√(1−t_ ^2 )) −1)) +(1/( (√(1−t^2 )) +1)) } (t/(2(√(1−t^2 )))){(((√(1−t)) −1−(√(1−t^2 )) −1)/((1−t^2 −1))} (1/2)×(t/( (√(1−t^2 )))){((−2)/(−t^2 ))} (1/(t(√(1−t^2 )))) so attached answer in right hand side ln(1−(√(1−t^2 )) ) is not correct
∫tdtt21−t2k2=1−t22kdk=−2tdt∫−kdk(1−k2)k∫dkk2−112∫(k+1)−(k−1)(k+1)(k−1)dk12∫dkk−1−12∫dkk+112{ln(k−1)−ln(k+1)}+c12{ln(k−1k+1)}+c12{ln(1−t2−11−t2+1)}+c←correctanswerrecheckofanswergivenRHSddt{ln(1−1−t2)}11−1−t2×0−121−t2×−2t111−1−t2×t1−t2nowrecheckresultobtainedddt[12{ln(1−t2−1)−ln(1−t2+1)}]12[{11−t2−1×121−t2×−2t}−{11−t2+1×−2t21−t2}]12[−t1−t2.(1−t2−1)+t1−t2.(1−t2+1)]t21−t2{−11−t2−1+11−t2+1}t21−t2{1−t−1−1−t2−1(1−t2−1}12×t1−t2{−2−t2}1t1−t2soattachedanswerinrighthandsideln(1−1−t2)isnotcorrect
Commented by arvinddayama01@gmail.com last updated on 06/Oct/18
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Answered by ajfour last updated on 04/Oct/18
let t=sin θ ∫((cos θdθ)/(sin θcos θ)) = ∫cosec θdθ =ln ∣cosec θ−cot θ∣+c =ln ∣(1/t)−((√(1−t^2 ))/t)∣+c = ln ∣1−(√(1−t^2 )) ∣−ln ∣t∣+c .
lett=sinθ∫cosθdθsinθcosθ=∫cosecθdθ=ln∣cosecθ−cotθ∣+c=ln∣1t−1−t2t∣+c=ln∣1−1−t2∣−ln∣t∣+c.
Commented by arvinddayama01@gmail.com last updated on 06/Oct/18
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