NCERT Class 12 · Ch 7 Exercise 7.7 · Q5
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.7 Question 5
Evaluate \(\int \sqrt{- x^{2} - 4 x + 1}\, dx\).
💡 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: \(\frac{\sqrt{5 - \left(x + 2\right)^{2}} \left(x + 2\right)}{2} + \frac{5 \operatorname{asin}{\left(\frac{\sqrt{5} \left(x + 2\right)}{5} \right)}}{2}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Complete the square 5−(x+2)².
- For J=∫√(u²+A)du, integrate by parts with dv=du. Use u²=(u²+A)−A to get 2J=u√(u²+A)+A∫du/√(u²+A). For √(A−u²), the analogous identity uses an arcsine. Substitute the standard log or arcsine primitive.
- An antiderivative is F(x)=\(\frac{\sqrt{5 - \left(x + 2\right)^{2}} \left(x + 2\right)}{2} + \frac{5 \operatorname{asin}{\left(\frac{\sqrt{5} \left(x + 2\right)}{5} \right)}}{2}\).
- Differentiate this expression to check that F′(x)=\(\sqrt{- x^{2} - 4 x + 1}\). Add an arbitrary constant C on each domain component.
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