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Class 12 Integrals · Long Answer3 Marksindefinite_integration

Class 12 Maths · Integrals · Long Answer #160

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

✅ Step-by-Step Solution

Final Answer: \(\frac{\log{\left(x^{2} + 1 \right)}}{5} + \frac{3 \log{\left(\left|{x + 2}\right| \right)}}{5} + \frac{\operatorname{atan}{\left(x \right)}}{5}\) + C
  1. Write the fraction as A/(x+2)+(Bx+C)/(x²+1).
  2. Equating coefficients gives A+B=1,2B+C=1,A+2C=1; solve A=3/5,B=2/5,C=1/5.
  3. Integrate the reciprocal linear term, the x/(x²+1) term and the 1/(x²+1) term separately.
  4. Therefore the required integral is \(\frac{\log{\left(x^{2} + 1 \right)}}{5} + \frac{3 \log{\left(\left|{x + 2}\right| \right)}}{5} + \frac{\operatorname{atan}{\left(x \right)}}{5}\) + C.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Write the fraction as A/(x+2)+(Bx+C)/(x²+1).
  • [1 mark] Equating coefficients gives A+B=1,2B+C=1,A+2C=1; solve A=3/5,B=2/5,C=1/5.
  • [1 mark] Integrate the reciprocal linear term, the x/(x²+1) term and the 1/(x²+1) term separately.
  • [0.5 marks] Therefore the required integral is \(\frac{\log{\left(x^{2} + 1 \right)}}{5} + \frac{3 \log{\left(\left|{x + 2}\right| \right)}}{5} + \frac{\operatorname{atan}{\left(x \right)}}{5}\) + C.

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