If equation \( \int \frac{d t}{\sqrt{3 a t-2 t^{2}}}=a^{x} \sin ^{-1}\left(\frac{t^{2}}{a^{2}}-1...

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If equation \( \int \frac{d t}{\sqrt{3 a t-2 t^{2}}}=a^{x} \sin ^{-1}\left(\frac{t^{2}}{a^{2}}-1\right) \), find the value of \( x \).
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