NCERT Class 12 · Ch 7 Exercise 7.4 · Q22
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.4 Question 22
Evaluate \(\int \frac{x + 3}{x^{2} - 2 x - 5}\, dx\).
💡 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: \(\frac{\sqrt{6} \log{\left(\left|{\frac{- x + 1 + \sqrt{6}}{x - 1 + \sqrt{6}}}\right| \right)}}{3} + \frac{\log{\left(\left|{- x^{2} + 2 x + 5}\right| \right)}}{2}\) + C.
- Work on each open interval where the integrand is real and defined.
- Write x+3=(1/2)(2x−2)+4, then use the log derivative and reciprocal difference-of-squares form.
- An antiderivative is F(x)=\(\frac{\sqrt{6} \log{\left(\left|{\frac{- x + 1 + \sqrt{6}}{x - 1 + \sqrt{6}}}\right| \right)}}{3} + \frac{\log{\left(\left|{- x^{2} + 2 x + 5}\right| \right)}}{2}\).
- Differentiate this expression to check that F′(x)=\(\frac{x + 3}{x^{2} - 2 x - 5}\). Add an arbitrary constant C on each domain component.
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