NCERT Class 12 · Ch 7 Exercise 7.4 · Q14
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.4 Question 14
Evaluate \(\int \frac{1}{- x^{2} + 3 x + 8}\, dx\).
💡 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: \(\frac{\sqrt{41} \log{\left(\left|{\frac{2 x - 3 + \sqrt{41}}{- 2 x + 3 + \sqrt{41}}}\right| \right)}}{41}\) + C.
- Work on each open interval where the integrand is real and defined.
- Write the denominator as 41/4−(x−3/2)² and use partial fractions.
- An antiderivative is F(x)=\(\frac{\sqrt{41} \log{\left(\left|{\frac{2 x - 3 + \sqrt{41}}{- 2 x + 3 + \sqrt{41}}}\right| \right)}}{41}\).
- Differentiate this expression to check that F′(x)=\(\frac{1}{- x^{2} + 3 x + 8}\). Add an arbitrary constant C on each domain component.
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