Class 12 Integrals · Short Answer2 Marksindefinite_integration

Class 12 Maths · Integrals · Short Answer #51

Evaluate \(\int \frac{x - 1}{\left(x - 3\right) \left(x - 2\right)}\,dx\). Work on each connected open interval where the real integrand is defined.

✅ Step-by-Step Solution

Final Answer: \(2 \log{\left(\left|{x - 3}\right| \right)} - \log{\left(\left|{x - 2}\right| \right)}\) + C

Decompose (x−1)/[(x−2)(x−3)]=A/(x−2)+B/(x−3).
Substitution at x=2 and x=3 gives A=−1,B=2. Integrating gives −ln|x−2|+2ln|x−3|+C.
Therefore the required integral is \(2 \log{\left(\left|{x - 3}\right| \right)} - \log{\left(\left|{x - 2}\right| \right)}\) + C.

[0.5 marks] Decompose (x−1)/[(x−2)(x−3)]=A/(x−2)+B/(x−3).
[1 mark] Substitution at x=2 and x=3 gives A=−1,B=2. Integrating gives −ln|x−2|+2ln|x−3|+C.
[0.5 marks] Therefore the required integral is \(2 \log{\left(\left|{x - 3}\right| \right)} - \log{\left(\left|{x - 2}\right| \right)}\) + C.

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