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Question

Use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.$\int _ { 1 } ^ { e } \frac { d y } { y \sqrt { 1 + ( \ln y ) ^ { 2 } } }$

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Answered 1 year ago
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First, do a u-substitution with $\color{#c34632}u=\ln y$ since $\ln y$ is the inside function. If $u=\ln y$, then $\color{#4257b2}du=\dfrac{1}{y}\;dy$. If $y=1$, then $u=\ln y=\ln 1=0$, and if $y=e$, then $u=\ln y=\ln e=1$. The lower bound after the substitution must then change to 0 and the upper bound must change to 1. Therefore:

\begin{align*} \int_1^e\frac{1}{y\sqrt{1+(\ln y)^2}}\;dy&=\int_1^e\dfrac{1}{\sqrt{(1+(\color{#c34632}\ln y\color{default})^2}}\cdot\color{#4257b2}\frac{1}{y}\;dy&&\text{Regroup.}\\ &=\int_0^1\frac{1}{\sqrt{1+\color{#c34632}u\color{default}^2}}\color{#4257b2}\;du&&\text{Substitute.} \end{align*}

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