Question: Evaluate \(\int \frac{dx}{\sqrt{25 - 16x^2}}\)

Step 1 (Formula / Governing Law): [0.25 Mark]
The integral is of the standard form \(\int \frac{dx}{\sqrt{a^2 - u^2}} = \sin^{-1}\left(\frac{u}{a}\right) + C\). To apply this, we must express the denominator in the form \(\sqrt{a^2 - (kx)^2}\).

Step 2 (Step-by-Step Mathematical Working): [0.5 Mark]
Given integral: \(I = \int \frac{dx}{\sqrt{25 - 16x^2}}\)

Factor out 16 from the square root to make the coefficient of \(x^2\) equal to 1:

$$I = \int \frac{dx}{\sqrt{16\left(\frac{25}{16} - x^2\right)}} = \int \frac{dx}{4\sqrt{\left(\frac{5}{4}\right)^2 - x^2}}$$

Now, apply the standard integration formula \(\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\left(\frac{x}{a}\right) + C\), where \(a = \frac{5}{4}\):

$$I = \frac{1}{4} \sin^{-1}\left(\frac{x}{5/4}\right) + C$$ $$I = \frac{1}{4} \sin^{-1}\left(\frac{4x}{5}\right) + C$$

Step 3 (Final Concluded Answer): [0.25 Mark]
Comparing the result with the given options, the correct choice is (C).

Final Answer: \(\frac{1}{4} \sin^{-1} \frac{4x}{5} + C\)

💡 EL10 Concept Insight: Think of this like "un-stretching" a spring; because the \(x\) is being multiplied by 4, the final answer needs to be divided by 4 to balance it out, just like how you divide by the derivative of the inside part when doing reverse-chain rule integration.