Question

How do you solve this question? Any help appreciated!

Answer 1

$\bf \displaystyle \int\limits_{1}^{e}\cfrac{1}{t}\cdot dt\\\\ -------------------------------\\\\ \textit{doing substitution}\\\\ u=\cfrac{1}{t}\implies u=t^{-1}\implies \cfrac{du}{dt}=-t^{-2}\implies \cfrac{du}{dt}=-\cfrac{1}{t^2}\\\\\\ -t^2du=dt\\\\ -------------------------------\\\\ \displaystyle \int\limits_{1}^{e}u\cdot -t^2du\impliedby \textit{now, let's do some substitution on the "t"}\\\\ -------------------------------$

$\bf u=\cfrac{1}{t}\implies t=\cfrac{1}{u}\implies t^2=\cfrac{1^2}{u^2}\implies t^2=\cfrac{1}{u^2}\\\\ -------------------------------\\\\ \displaystyle \int\limits_{1}^{e}u\cdot -\cfrac{1}{u^2}\cdot du\implies -1\int\limits_{1}^{e}\cfrac{1}{u}\cdot du\implies \left. -ln|u| \cfrac{}{}\right]_1^e$

$\bf \left. -ln\left( \frac{1}{t} \right) \cfrac{}{}\right]_1^e\implies \left[ -ln\left( \frac{1}{e} \right) \right]-\left[ -ln\left( \frac{1}{1} \right) \right]\implies \left[ -ln\left( e^{-1}\right) \right]-\left[ -ln\left( 1\right) \right] \\\\\\\ [-(-1)]-[-(0)]\implies 1-0\implies 1$