Indefinite Integration
Indefinite Integration
nta_pyq_2025_jan
Grade 12
Question:
If \int 2 2x +5x+9 dx = x\sqrtx 2 + x + 1 + \alpha\sqrtx 2 + x + 1+ \beta log ∣ e∣ x + 1 2 + \sqrtx 2 + x + 1∣ ∣ + C , where C is the \sqrtx2 +x+1 constant of integration, then \alpha + 2\beta is equal to _______.
Step-by-Step Solution
Key Concept: Apply the core result for integration by substitution and identities and simplify using the given constraints.
2x + 5x + 9 = A (x + x + 1) + B(2x + 1) + C 2 2 (16) A = 2 B = 3 C = 11 2 2 3 2x + 1 11 dx 2 2\int \sqrtx + x + 1dx + \int dx + \int 2 \sqrtx2 + x + 1 2 \sqrtx2 + x + 1 2 2 \sqrt3 1 11 dx 2 2\int (x + ) + ( ) dx + 3\sqrtx + x + 1 + \int ⎷ 2 2 2 2 2 1 \sqrt3 \sqrt (x + ) + ( ) 2 2 1 ⎛ x + 2 3 1 ⎞ 2 2 2 2 \sqrtx + x + 1 + ln(x + + \sqrtx + x + 1) + 3\sqrt x + x + 1 ⎝ 2 8 2 ⎠ 11 1 2 + ℓn (x + + \sqrtx + x + 1) + C 2 2 7 25 \alpha = \beta = 2 4 \alpha + 2\beta = 16
Correct Answer: 16