NCERT Class 12 · Ch 7 Exercise 7.4 · Q20
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.4 Question 20
Evaluate \(\int \frac{x + 2}{\sqrt{- x^{2} + 4 x}}\, dx\).
💡 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: \(- \sqrt{- x^{2} + 4 x} + 4 \operatorname{asin}{\left(\frac{x}{2} - 1 \right)}\) + C.
- Work on each open interval where the integrand is real and defined.
- Write x+2=(x−2)+4; complete the square 4−(x−2)².
- An antiderivative is F(x)=\(- \sqrt{- x^{2} + 4 x} + 4 \operatorname{asin}{\left(\frac{x}{2} - 1 \right)}\).
- Differentiate this expression to check that F′(x)=\(\frac{x + 2}{\sqrt{- x^{2} + 4 x}}\). Add an arbitrary constant C on each domain component.
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