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CBSE Class 12 Mathematics 2026 Supplementary Paper 65(B)/7 · Q283 Markssubstitution

CBSE Class 12 Mathematics 2026 Supplementary Paper 65(B)/7 · Question 28 (Integrals)

Evaluate \(\int e^x\,dx/\sqrt{5-4e^x-e^{2x}}\). OR Evaluate \(\int x^2\,dx/[(x^2+4)(x^2+9)]\).
Alternative 1 3 Marks
Evaluate \(\int e^x\,dx/\sqrt{5-4e^x-e^{2x}}\).
Answer: arcsin((eˣ+2)/3)+C on its real domain.
  1. Set u=eˣ, so du=eˣ dx.
  2. Complete the square: 5−4u−u²=9−(u+2)².
  3. Use the arcsine primitive and substitute u=eˣ.
Alternative 2 3 Marks
Evaluate \(\int x^2\,dx/[(x^2+4)(x^2+9)]\).
Answer: −(2/5)arctan(x/2)+(3/5)arctan(x/3)+C.
  1. Write the fraction as A/(x²+4)+B/(x²+9).
  2. Matching terms gives A=−4/5 and B=9/5.
  3. Integrate with the two arctangent primitives.

✅ Step-by-Step Solution

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All Questions in CBSE Class 12 Mathematics 2026 Supplementary Paper 65(B)/7

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