Question

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

• A. $log\left|\frac{2logx+1}{3logx+2}\right|+C$
• B. $\frac{1}{2}log\left|\frac{2logx+1}{3logx+2}\right|+C$
• C. $log\left|\frac{3logx+2}{2logx+1}\right|+C$
• 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 $logx=t\Rightarrow \frac{1}{x}dx=dt$

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

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

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

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