integrate dx/x(6(logx) ^2 +7logx +2)

Asked by sidhusoni115 | 3rd Oct, 2020, 08:08: AM

Expert Answer:

TO integrate dx/x(6(log x)2 + 7 logx + 2)
straight I equals integral fraction numerator dx over denominator straight x open square brackets 6 open parentheses logx close parentheses squared plus 7 logx plus 2 close square brackets end fraction
Take space logx equals straight y rightwards double arrow straight x equals straight e to the power of straight y rightwards double arrow dx equals straight e to the power of straight y dy
rightwards double arrow straight I equals integral fraction numerator straight e to the power of straight y dy over denominator straight e to the power of straight y open square brackets 6 straight y squared plus 7 straight y plus 2 close square brackets end fraction equals integral fraction numerator dy over denominator open square brackets 6 straight y squared plus 7 straight y plus 2 close square brackets end fraction equals integral fraction numerator dy over denominator open parentheses 3 straight y plus 2 close parentheses open parentheses 2 straight y plus 1 close parentheses end fraction
rightwards double arrow straight I equals integral open square brackets fraction numerator 2 over denominator 2 straight y plus 1 end fraction minus fraction numerator 3 over denominator 3 straight y plus 2 end fraction close square brackets dy equals integral fraction numerator 2 over denominator 2 straight y plus 1 end fraction dy minus integral fraction numerator 3 over denominator 3 straight y plus 2 end fraction dy
rightwards double arrow straight I equals log open parentheses 2 straight y plus 1 close parentheses minus log open parentheses 3 straight y plus 2 close parentheses
rightwards double arrow straight I equals log open parentheses 2 logx plus 1 close parentheses minus log open parentheses 3 logx plus 2 close parentheses
Hence comma space integral fraction numerator dx over denominator straight x open square brackets 6 open parentheses logx close parentheses squared plus 7 logx plus 2 close square brackets end fraction equals log open parentheses 2 logx plus 1 close parentheses minus log open parentheses 3 logx plus 2 close parentheses equals log open parentheses fraction numerator 2 logx plus 1 over denominator 3 logx plus 2 end fraction close parentheses

Answered by Renu Varma | 5th Oct, 2020, 10:55: AM