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The value of \(x\) that satisfies the equation \(\int_{\sqrt{2}}^x \frac{d t}{|t| \sqrt{t^2-1}}=\frac{\pi}{12}\) is
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\(\begin{aligned}
& \int_{\sqrt{2}}^x \frac{d t}{|t| \sqrt{t^2-1}}=\frac{\pi}{12} \\
& \Rightarrow \quad\left[\sec ^{-1} t\right]_{\sqrt{2}}^x=\frac{\pi}{12} \Rightarrow \sec ^{-1} x-\sec ^{-1} \sqrt{2}=\frac{\pi}{12} \\
& \Rightarrow \quad \sec ^{-1} x=\frac{\pi}{12}+\frac{\pi}{4}=\frac{\pi}{3} \\
& \Rightarrow \quad x=\sec \left(\frac{\pi}{3}\right)=2
\end{aligned}\)
& \int_{\sqrt{2}}^x \frac{d t}{|t| \sqrt{t^2-1}}=\frac{\pi}{12} \\
& \Rightarrow \quad\left[\sec ^{-1} t\right]_{\sqrt{2}}^x=\frac{\pi}{12} \Rightarrow \sec ^{-1} x-\sec ^{-1} \sqrt{2}=\frac{\pi}{12} \\
& \Rightarrow \quad \sec ^{-1} x=\frac{\pi}{12}+\frac{\pi}{4}=\frac{\pi}{3} \\
& \Rightarrow \quad x=\sec \left(\frac{\pi}{3}\right)=2
\end{aligned}\)
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