NCERT Class 12 · Ch 7 Exercise 7.6 · Q19
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.6 Question 19
Evaluate \(\int \left(\frac{1}{x} - \frac{1}{x^{2}}\right) e^{x}\, dx\).
💡 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: \(\frac{e^{x}}{x}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Recognize eˣ[g(x)+g′(x)] with g=\(\frac{1}{x}\) and g′=\(- \frac{1}{x^{2}}\).
- The product rule makes its primitive eˣg(x); substitution into the original integrand verifies the match.
- An antiderivative is F(x)=\(\frac{e^{x}}{x}\).
- Differentiate this expression to check that F′(x)=\(\left(\frac{1}{x} - \frac{1}{x^{2}}\right) e^{x}\). Add an arbitrary constant C on each domain component.
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