NCERT Class 12 · Ch 7 Miscellaneous Exercise · Q6
NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 6
Evaluate \(\int \frac{5 x}{\left(x + 1\right) \left(x^{2} + 9\right)}\, dx\).
💡 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: \(\frac{\log{\left(x^{2} + 9 \right)}}{4} - \frac{\log{\left(\left|{x + 1}\right| \right)}}{2} + \frac{3 \operatorname{atan}{\left(\frac{x}{3} \right)}}{2}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Use partial fractions. The decomposition is \(\frac{x + 9}{2 \left(x^{2} + 9\right)} - \frac{1}{2 \left(x + 1\right)}\).
- An antiderivative is F(x)=\(\frac{\log{\left(x^{2} + 9 \right)}}{4} - \frac{\log{\left(\left|{x + 1}\right| \right)}}{2} + \frac{3 \operatorname{atan}{\left(\frac{x}{3} \right)}}{2}\).
- Differentiate this expression to check that F′(x)=\(\frac{5 x}{\left(x + 1\right) \left(x^{2} + 9\right)}\). Add an arbitrary constant C on each domain component.
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