Evaluate \[ \int\frac{dx}{x(\ln x)^2}, \qquad \int_2^\infty\frac{dx}{x(\ln x)^2}, \] and \[ \int \frac1{x^2}\cos\!\left(\frac1x\right)\,dx. \]
Solution
For the first integral, let \(u=\ln x\), so \(du=dx/x\): \[ \int\frac{dx}{x(\ln x)^2} =\int u^{-2}\,du =\boxed{-\frac1{\ln x}+C}. \] Thus \[ \int_2^\infty\frac{dx}{x(\ln x)^2} =\lim_{R\to\infty}\left[-\frac1{\ln x}\right]_{2}^{R} =\boxed{\frac1{\ln2}}. \] For the second antiderivative, let \(u=1/x\), so \(du=-dx/x^2\): \[ \boxed{\int \frac1{x^2}\cos\!\left(\frac1x\right)\,dx=-\sin\!\left(\frac1x\right)+C.} \]