CBSE Class 12 Mathematics 2026 Main Paper 65(B) · Q273 Markssubstitution
CBSE Class 12 Mathematics 2026 Main Paper 65(B) · Question 27 (Integrals)
Evaluate \(\int x\,dx/\sqrt{7-6x-x^2}\). OR Evaluate \(\int 2\,dx/[(1-x)(1+x^2)]\).
Alternative 1 3 Marks
Evaluate \(\int x\,dx/\sqrt{7-6x-x^2}\).
Answer: −√(7−6x−x²)−3 arcsin((x+3)/4)+C.
- Write the radicand as 16−(x+3)² and x=(x+3)−3.
- Integrate the first numerator term as −√(16−(x+3)²).
- The remaining term gives −3 arcsin((x+3)/4).
Alternative 2 3 Marks
Evaluate \(\int 2\,dx/[(1-x)(1+x^2)]\).
Answer: −ln|1−x|+(1/2)ln(1+x²)+arctan x+C.
- Decompose as 1/(1−x)+(x+1)/(1+x²).
- Integrate the three numerator terms using logarithms and arctangent.
- Add the integration constant on each component of the domain.