NCERT Class 12 · Ch 7 Exercise 7.6 · Q20
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.6 Question 20
Evaluate \(\int \frac{\left(x - 3\right) e^{x}}{\left(x - 1\right)^{3}}\, dx\).
💡 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: \(\frac{e^{x}}{\left(x - 1\right)^{2}}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Recognize eˣ[g(x)+g′(x)] with g=\(\frac{1}{\left(x - 1\right)^{2}}\) and g′=\(- \frac{2}{\left(x - 1\right)^{3}}\).
- The product rule makes its primitive eˣg(x); substitution into the original integrand verifies the match.
- An antiderivative is F(x)=\(\frac{e^{x}}{\left(x - 1\right)^{2}}\).
- Differentiate this expression to check that F′(x)=\(\frac{\left(x - 3\right) e^{x}}{\left(x - 1\right)^{3}}\). Add an arbitrary constant C on each domain component.
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