Question

Evaluate: $\int\frac{1}{x\left\{6\left(log\,x\right)^{2}+7log\,x +2\right\}}dx$

• A. $log\left|\frac{2\,log\,x+1}{3\,log \,x+2}\right|+C$
• B. $\frac{1}{2} log\left|\frac{2\, log\,x+1}{3\, log \,x+2}\right|+C$
• C. $log\left|\frac{3\, log\,x+2}{2\,log\, x+1}\right|+C$
• D. None of these

Answer 1

Answer: (A)

Let $I=\int\frac{1}{x\left\{6\left(logx\right)^{2}+7logx+2\right\}}dx $

Let $logx =t \Rightarrow \frac{1}{x}dx = dt$

$\therefore I = \int\frac{1}{6t^{2}+7t +2}dt$

$\Rightarrow I =\frac{1}{6}\int\frac{1}{\left(t+\frac{7}{12}\right)^{2}-\left(\frac{1}{12}^{2}\right)}dt$

$ \Rightarrow I =\frac{1}{6}\times\frac{1}{2\left(\frac{1}{12}\right)}log\left|\frac{t+\frac{7}{12}-\frac{1}{12}}{t+\frac{7}{12}+\frac{1}{12}}\right| +K$

$ \Rightarrow I =log\left|\frac{2t+1}{3t+2}\right|+ C = log \left|\frac{2log x+1}{3log x +2}\right|+C \,\,\,\ \left[where, C = K +log \frac{3}{2}\right]$