Class 12 Integrals · Short Answer2 Marksindefinite_integration
Class 12 Maths · Integrals · Short Answer #48
Evaluate \(\int \frac{x}{x^{2} + 3 x + 2}\,dx\). Work on each connected open interval where the real integrand is defined.
✅ Step-by-Step Solution
Final Answer: \(- \log{\left(\left|{x + 1}\right| \right)} + 2 \log{\left(\left|{x + 2}\right| \right)}\) + C
- Factor the denominator as (x+1)(x+2).
- Partial fractions give x/[(x+1)(x+2)]=−1/(x+1)+2/(x+2). Integrate each reciprocal.
- Therefore the required integral is \(- \log{\left(\left|{x + 1}\right| \right)} + 2 \log{\left(\left|{x + 2}\right| \right)}\) + C.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Factor the denominator as (x+1)(x+2).
- [1 mark] Partial fractions give x/[(x+1)(x+2)]=−1/(x+1)+2/(x+2). Integrate each reciprocal.
- [0.5 marks] Therefore the required integral is \(- \log{\left(\left|{x + 1}\right| \right)} + 2 \log{\left(\left|{x + 2}\right| \right)}\) + C.
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