The given integral is of the form ∫ dx / √(a² - u²), which evaluates to sin⁻¹(u/a) + C. We need to manipulate the denominator to match this form.
Factor out 16 from the square root in the denominator: $$ \int \frac{dx}{\sqrt{25 - 16x^2}} = \int \frac{dx}{\sqrt{16(\frac{25}{16} - x^2)}} $$ $$ = \frac{1}{4} \int \frac{dx}{\sqrt{(\frac{5}{4})^2 - x^2}} $$
Using the standard formula ∫ dx / √(a² - x²) = sin⁻¹(x/a) + C, where a = 5/4: $$ = \frac{1}{4} \sin^{-1}\left(\frac{x}{5/4}\right) + C $$ $$ = \frac{1}{4} \sin^{-1}\left(\frac{4x}{5}\right) + C $$
Final Answer: Option (C)
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