NCERT Class 12 · Ch 7 Exercise 7.4 · Q18
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.4 Question 18
Evaluate \(\int \frac{5 x - 2}{3 x^{2} + 2 x + 1}\, dx\).
💡 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: \(\frac{5 \log{\left(\left|{3 x^{2} + 2 x + 1}\right| \right)}}{6} - \frac{11 \sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \left(3 x + 1\right)}{2} \right)}}{6}\) + C.
- Work on each open interval where the integrand is real and defined.
- Write 5x−2=(5/6)(6x+2)−11/3. For the remainder, complete the square 3x²+2x+1=[(3x+1)²+2]/3.
- An antiderivative is F(x)=\(\frac{5 \log{\left(\left|{3 x^{2} + 2 x + 1}\right| \right)}}{6} - \frac{11 \sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \left(3 x + 1\right)}{2} \right)}}{6}\).
- Differentiate this expression to check that F′(x)=\(\frac{5 x - 2}{3 x^{2} + 2 x + 1}\). Add an arbitrary constant C on each domain component.
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