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Class 12 Integrals · Long Answer3 Marksindefinite_integration

Class 12 Maths · Integrals · Long Answer #76

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

✅ Step-by-Step Solution

Final Answer: \(x - 2 \log{\left(\left|{\frac{x - 2}{x - 1}}\right| \right)}\) + C
  1. The numerator is the denominator minus 2, so the integrand is 1−2/[(x−1)(x−2)].
  2. Since 1/[(x−1)(x−2)]=−1/(x−1)+1/(x−2), integrate the linear reciprocals.
  3. The result before combining logs is x+2ln|x−1|−2ln|x−2|.
  4. Therefore the required integral is \(x - 2 \log{\left(\left|{\frac{x - 2}{x - 1}}\right| \right)}\) + C.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] The numerator is the denominator minus 2, so the integrand is 1−2/[(x−1)(x−2)].
  • [1 mark] Since 1/[(x−1)(x−2)]=−1/(x−1)+1/(x−2), integrate the linear reciprocals.
  • [1 mark] The result before combining logs is x+2ln|x−1|−2ln|x−2|.
  • [0.5 marks] Therefore the required integral is \(x - 2 \log{\left(\left|{\frac{x - 2}{x - 1}}\right| \right)}\) + C.

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