5.5 The Substitution Rule/59

$\int _{1}^{2}{\frac {e^{\frac {1}{x}}}{x^{2}}}\,dx$

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}u&={\frac {1}{x}}\\[2ex]du&=-{\frac {1}{x^{2}}}dx\\[2ex]-du&={\frac {1}{x^{2}}}dx\\[2ex]\end{aligned}}}

New upper limit:${\frac {1}{2}}={\frac {1}{(2)}}$
New lower limit: Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 1={\frac {1}{(1)}}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \int_{1}^{2}\frac{ e^\frac{1}{x}}{x^2}\,dx &=\int_{1}^{2} e^\frac{1}{x}(\frac{1}{x^2}\,dx) &=\int_{1}^{\frac{1}{2}}e^u\,(-du) \\[2ex] &=-\int_{1}^{\frac{1}{2}}e^u\,du \\[2ex] &=-e^u\bigg|_{1}^{\frac{1}{2}} \\[2ex] &=-\sqrt{e} - (-e^1) \\[2ex] &=e-\sqrt{e} \end {align} }