Integral Calculus
86402
The value of \(x\) that satisfies the equation \(\int_{\sqrt{2}}^{x} \frac{d t}{|t| \sqrt{t^{2}-1}}=\frac{\pi}{12}\) is
1 1
2 0
3 \(-\sqrt{2}\)
4 2
Explanation:
(D) : \(\int_{\sqrt{2}}^{\mathrm{x}} \frac{\mathrm{dt}}{|\mathrm{t}| \sqrt{\mathrm{t}^{2}-1}}=\frac{\pi}{12}\)
\(\left[\sec ^{-1} t\right]_{\sqrt{2}}^{x}=\frac{\pi}{12} \Rightarrow \sec ^{-1} \mathrm{x}-\sec ^{-1} \sqrt{2}=\frac{\pi}{12}\)
\(\Rightarrow \sec ^{-1} \mathrm{x}=\frac{\pi}{12}+\frac{\pi}{4}=\frac{\pi}{3}\)
\(\Rightarrow \mathrm{x}=\sec \left(\frac{\pi}{3}\right)=2\)