Question

If
$\displaystyle\int\limits^{\pi/3}_0 { \frac{ \tan \theta}{ \sqrt{2k \sec \theta} } } \: d \theta = 1 - \frac{1}{ \sqrt{2} } ,(k > 0)$
Then Find the value of k

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Answer 1

$\texttt{\textsf{\huge{\underline{★Given}:}}}$

$\displaystyle\int\limits^{\ \frac{\pi}{3} }_0 { \frac{ \tan \theta}{ \sqrt{2k \sec \theta} } } \: d \theta = 1 - \frac{1}{ \sqrt{2} } ,(k > 0)$

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$\texttt{\textsf{\huge{\underline{★To Find}:}}}$

$\texttt{\textsf{☼︎The \: value \: of \: k}}$

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$\texttt{\textsf{\huge{\underline{Solution}:}}}$

$LHS = \frac{1}{ \sqrt{2k} } \displaystyle \int\limits_{0}^{ \frac{\pi}{3} }</p><p>\: \: \frac{tanθ}{ \sqrt{secθ} } \: \: dθ$

$= \frac{1}{ \sqrt{2k} } \displaystyle \int\limits_{0}^{ \frac{\pi}{3} }</p><p>\: \: \frac{sinθ}{ \sqrt{cosθ} } \: \: dθ$

$= \frac{ - 1}{ \sqrt{2k} } ∣2 \sqrt{cosθ} | \: 0 \: \frac{\pi}{3} </p><p></p><p>$

$= - \frac{ \sqrt{2} }{ \sqrt{k} } ( \frac{1}{ \sqrt{2} } - 1)$

$= \frac{ \sqrt{2} }{ \sqrt{k} } ( 1 - \frac{1}{ \sqrt{2} } )$

$RHS = 1 - \frac{1}{ \sqrt{2} }$

$∴ \: \: k = 2$

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Answer 2

Answer

$\texttt{\textsf{\huge{\underline{★Given}:}}}$

$\begin{gathered} \displaystyle\int\limits^{\ \frac{\pi}{3} }_0 { \frac{ \tan \theta}{ \sqrt{2k \sec \theta} } } \: d \theta = 1 - \frac{1}{ \sqrt{2} } ,(k > 0) \\ \end{gathered}0∫ 3π2ksecθtanθdθ=1−21,(k>0)$

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$\texttt{\textsf{\huge{\underline{★To Find}:}}}$

$\texttt{\textsf{☼︎The \: value \: of \: k}}$

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$\texttt{\textsf{\huge{\underline{Solution}:}}}$

$LHS = \frac{1}{ \sqrt{2k} } \displaystyle \int\limits_{0}^{ \frac{\pi}{3} } < /p > < p > \: \: \frac{tanθ}{ \sqrt{secθ} } \: \: dθLHS=2k10∫3π</p><p>secθtanθdθ$

$= \frac{1}{ \sqrt{2k} } \displaystyle \int\limits_{0}^{ \frac{\pi}{3} } < /p > < p > \: \: \frac{sinθ}{ \sqrt{cosθ} } \: \: dθ=2k10∫3π</p><p>cosθsinθdθ$

$= \frac{ - 1}{ \sqrt{2k} } ∣2 \sqrt{cosθ} | \: 0 \: \frac{\pi}{3} < /p > < p > < /p > < p >=2k−1∣2cosθ∣03π</p><p></p><p>$

$\begin{gathered} = - \frac{ \sqrt{2} }{ \sqrt{k} } ( \frac{1}{ \sqrt{2} } - 1) \\ \end{gathered}=−k2(21−1)$

$</p><p>\begin{gathered} = \frac{ \sqrt{2} }{ \sqrt{k} } ( 1 - \frac{1}{ \sqrt{2} } ) \\ \end{gathered}=k2(1−21)</p><p>\begin{gathered}RHS = 1 - \frac{1}{ \sqrt{2} } \\ \end{gathered}RHS=1−21$

$∴ \: \: k = 2∴k=2$

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