NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.8 Question 19

✅ Step-by-Step Solution

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)=\(3 \log{\left(x^{2} + 4 \right)} + \frac{3 \operatorname{atan}{\left(\frac{x}{2} \right)}}{2}\).
4. At the upper and lower limits, F(b)=\(\frac{3 \pi}{8} + 3 \log{\left(8 \right)}\) and F(a)=\(3 \log{\left(4 \right)}\).
5. By the fundamental theorem (using one-sided limits for any integrable endpoint), I=F(b)−F(a)=\(\frac{3 \pi}{8} + \log{\left(8 \right)}\).