Mathbee CBSE Mathbee · CBSE Class 12 Maths
Class 12 Integrals · Very Long Answer5 Marksindefinite_integration

Class 12 Maths · Integrals · Very Long Answer #108

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

✅ Step-by-Step Solution

Final Answer: \(\frac{\log{\left(\frac{x^{2} + 1}{x^{2} + 3} \right)}}{2}\) + C
  1. Set u=x², du=2x dx.
  2. The integral becomes ∫du/[(u+1)(u+3)]. Set this fraction equal to A/(u+1)+B/(u+3).
  3. Equate 1=(A+B)u+3A+B. Thus A+B=0 and 3A+B=1, giving A=1/2,B=−1/2.
  4. Integrate to (1/2)ln|u+1|−(1/2)ln|u+3| and substitute u=x²; both arguments are positive.
  5. Therefore the required integral is \(\frac{\log{\left(\frac{x^{2} + 1}{x^{2} + 3} \right)}}{2}\) + C.

📋 CBSE Stepwise Marking Scheme

  • [1 mark] Set u=x², du=2x dx.
  • [1 mark] The integral becomes ∫du/[(u+1)(u+3)]. Set this fraction equal to A/(u+1)+B/(u+3).
  • [1 mark] Equate 1=(A+B)u+3A+B. Thus A+B=0 and 3A+B=1, giving A=1/2,B=−1/2.
  • [1 mark] Integrate to (1/2)ln|u+1|−(1/2)ln|u+3| and substitute u=x²; both arguments are positive.
  • [1 mark] Therefore the required integral is \(\frac{\log{\left(\frac{x^{2} + 1}{x^{2} + 3} \right)}}{2}\) + C.

Master Chapter 7: Integrals

Practise interactive questions, flashcards, visual mind maps, and timed board tests for Integrals.
📘 Chapter Notes & All Exercises 🚀 Open Interactive Workspace →
← Previous Question
Evaluate \(\int \frac{5 x + 3}{\sqrt{x^{2} + 4 x + 10}}\,dx\). Work…
Next Question →
Evaluate \(\int \frac{x + 2}{\sqrt{x^{2} + 5 x + 6}}\,dx\). Work on…

More Practice Questions in Integrals

Q1Q2Q3Q4Q5Q6Q7Q8Q9Q10Q11Q12Q13Q14Q15Q16Q17Q18Q19Q20Q21Q22Q23Q24Q25Q26Q27Q28Q29Q30Q31Q32Q33Q34Q35Q36Q37Q38Q39Q40

Related Questions in Integrals

Class 12 Integrals · Mcq Single Correct
Evaluate \(\int x^2e^{x^3}\,dx\).
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int e^x(1+x)\sec^2(xe^x)\,dx\) on an interval where defined.
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int e^x(\cos x-\sin x)\,dx\).
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int dx/(\sin^2x\cos^2x)\) on an interval avoiding its poles.
Class 12 Integrals · Mcq Single Correct
If \(\int\frac{3e^x-5e^{-x}}{4e^x+5e^{-x}}dx=ax+b\ln|4e^x+5e^{-x}|+C\), find a,b.
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int_{-1}^{1}|x-2|/(x-2)\,dx\).