NCERT Class 12 · Ch 7 Miscellaneous Exercise · Q3
NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 3
Evaluate \(\int \frac{1}{x \sqrt{a x - x^{2}}}\, dx\).
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: \(- \frac{2 \sqrt{a x - x^{2}}}{a x}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Put t=a/x, or differentiate √(ax−x²)/x. This requires a≠0 and a real positive radicand.
- An antiderivative is F(x)=\(- \frac{2 \sqrt{a x - x^{2}}}{a x}\).
- Differentiate this expression to check that F′(x)=\(\frac{1}{x \sqrt{a x - x^{2}}}\). Add an arbitrary constant C on each domain component.
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