I was solving a integration problem in which we have to integrate the following $$\int\frac{dx}{x\sqrt{x^2+1}}$$
I tried it lot , I think almost got the last step but the answer did not match. My try and the answer are as follows $$x=\frac1t,\quad dx=-\frac{dt}{t^2},$$ \begin{align} \int\frac{dx}{x\sqrt{x^2+1}}&=-\int\frac{dt}{\sqrt{1+4t^2}}\\ &=-\frac12\int\frac{dt}{\sqrt{\frac14+t^2}}\\ &=-\ln\left[t+\sqrt{\frac14+t^2}\right]+c \end{align} and the answer is $$\ln\left[\left(x+\frac12\right)+\sqrt{x^2+x+1}\right]+c.$$