NCERT Class 12 · Ch 7 Exercise 7.4 · Q11
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.4 Question 11
Evaluate \(\int \frac{1}{9 x^{2} + 6 x + 5}\, dx\).
💡 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: \(\frac{\operatorname{atan}{\left(\frac{3 x}{2} + \frac{1}{2} \right)}}{6}\) + C.
- Work on each open interval where the integrand is real and defined.
- Complete the square: (3x+1)²+4; put u=(3x+1)/2.
- An antiderivative is F(x)=\(\frac{\operatorname{atan}{\left(\frac{3 x}{2} + \frac{1}{2} \right)}}{6}\).
- Differentiate this expression to check that F′(x)=\(\frac{1}{9 x^{2} + 6 x + 5}\). Add an arbitrary constant C on each domain component.
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