NCERT Class 12 · Ch 7 Exercise 7.9 · Q8
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.9 Question 8
Evaluate \(\int\limits_{1}^{2} \left(\frac{1}{x} - \frac{1}{2 x^{2}}\right) e^{2 x}\, dx\).
💡 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: \(\frac{\left(-2 + e^{2}\right) e^{2}}{4}\)
- Work on each open interval where the original integrand is real and defined.
- Recognize d[e^(2x)/(2x)]/dx by the product rule.
- An antiderivative is F(x)=\(\frac{e^{2 x}}{2 x}\).
- At the upper and lower limits, F(b)=\(\frac{e^{4}}{4}\) and F(a)=\(\frac{e^{2}}{2}\).
- By the fundamental theorem (using one-sided limits for any integrable endpoint), I=F(b)−F(a)=\(\frac{\left(-2 + e^{2}\right) e^{2}}{4}\).
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