CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q313 Marksintegrals
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Question 31 (Integrals)
Find: ∫ √((x+2)/(x−2)) dx. OR Find: ∫ x²/((x²+9) (x²+16)) dx.
Alternative 1 3 Marks
Find: ∫ √((x+2)/(x−2)) dx.
Answer: For x>2: F(x)+C. For x<−2: −F(x)+C, where F(x)=√(x²−4)+2 ln abs(x+√(x²−4)).
- The integrand is real for x≤−2 or x>2. On the open components the square-root quotient equals sign(x+2)(x+2)/√(x²−4).
- The derivative of F(x)=√(x²−4)+2 ln abs(x+√(x²−4)) is (x+2)/√(x²−4). The sign factor is +1 for x>2 and −1 for x<−2.
- Use F+C on the positive component and −F+C on the negative component, allowing independent constants.
Alternative 2 3 Marks
Find: ∫ x²/((x²+9) (x²+16)) dx.
Answer: −(3/7)tan⁻¹(x/3)+(4/7)tan⁻¹(x/4)+C.
- Decompose the rational function directly as A/(x²+9)+B/(x²+16).
- Matching coefficients gives A+B=1 and 16A+9B=0, so A=−9/7 and B=16/7.
- Integrate each term with the scaled-arctangent formula, giving −(3/7)tan⁻¹(x/3)+(4/7)tan⁻¹(x/4)+C.