NCERT Class 12 · Ch 7 Exercise 7.5 · Q22
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.5 Question 22
Evaluate \(\int \frac{x}{\left(x - 2\right) \left(x - 1\right)}\, dx\).
💡 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: B: \(2 \log{\left(\left|{x - 2}\right| \right)} - \log{\left(\left|{x - 1}\right| \right)}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Partial fractions: x/[(x−1)(x−2)]=−1/(x−1)+2/(x−2).
- An antiderivative is F(x)=\(2 \log{\left(\left|{x - 2}\right| \right)} - \log{\left(\left|{x - 1}\right| \right)}\).
- Differentiate this expression to check that F′(x)=\(\frac{x}{\left(x - 2\right) \left(x - 1\right)}\). Add an arbitrary constant C on each domain component.
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