Discussion
None of these four is a valid substitution for \( \int \sqrt{a^{2} - x^{2}}\,dx \) except
Answer: B
With \( x = a\sin\theta \) the radical becomes \( a\cos\theta \), removing the root cleanly. The tangent substitution suits \( a^{2} + x^{2} \), and the secant and hyperbolic cosine suit \( x^{2} - a^{2} \); each of those three leaves the wrong sign under the root here and produces an imaginary result
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