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NCERT Class 12 · Ch 7 Exercise 7.8 · Q16

NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.8 Question 16

Evaluate \(\int\limits_{1}^{2} \frac{5 x}{x^{2} + 4 x + 3}\, dx\).

💡 Concept Hints

  1. Identify the domain and the inner and outer functions, or the relevant one-sided limits.
  2. Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.

✅ Step-by-Step Solution

Final Answer: \(\log{\left(\frac{78125 \sqrt{30}}{221184} \right)}\)
  1. Work on each open interval where the original integrand is real and defined.
  2. Use linearity and the elementary power, logarithm, trigonometric and exponential rules. Where a product x eˣ occurs use integration by parts; for rational terms split off the denominator derivative.
  3. An antiderivative is F(x)=\(- \frac{5 \log{\left(\left|{x + 1}\right| \right)}}{2} + \frac{15 \log{\left(\left|{x + 3}\right| \right)}}{2}\).
  4. At the upper and lower limits, F(b)=\(- \frac{5 \log{\left(3 \right)}}{2} + \frac{15 \log{\left(5 \right)}}{2}\) and F(a)=\(- \frac{5 \log{\left(2 \right)}}{2} + \frac{15 \log{\left(4 \right)}}{2}\).
  5. By the fundamental theorem (using one-sided limits for any integrable endpoint), I=F(b)−F(a)=\(\log{\left(\frac{78125 \sqrt{30}}{221184} \right)}\).

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