Question

Evaluate:

$\underset{2}{\overset{e}{\int }}\left(\frac{1}{\ln x}-\frac{1}{\ln x^2}\right)dx$

Answer 1

$\begin{array}{c}\underset{2}{\overset{e}{\int }}\left(\frac{1}{\ln x}-\frac{1}{\ln x^2}\right)dx\\ \ln x=t,\ln e^2=t\\ \frac{1}{x}dx=dt\\ dx=xdt\left(x=e^t\right)\\ \underset{2}{\overset{1}{\int }}{\left(\frac{1}{t}-\frac{1}{t^2}\right)}^{e^{t}dt}dx\\ \underset{{\ln }^2}{\overset{1}{\int }}\left(\frac{1^{et}}{t}-\frac{e^t}{t^2}\right)dt\\ =\int e^x\left[f\left(x\right)+f^{\prime }\left(x\right)\right]=e^x\left(fx\right)\\ ={{\left[e^t\times \frac{1}{t}\right]}^1}_{{\ln }^2}\\ \left[e-e^{{\ln }^2}\times \frac{1}{{\ln }^2}\right]\\ =\left[e-\frac{2}{{\ln }^2}\right]\end{array}$