Project ForgeEngineering Mathematics
Calculus
GATE 2022
Medium
Given ∫₋∞^∞ e^(−x²) dx = √π. If a and b are positive integers, the value of ∫₋∞^∞ e^(−a(x+b)²) dx is ________.
- A√(πa)
- √(π/a)
- Cb√(πa)
- Db√(π/a)
Explanation
Substitute u=√a(x+b), so du=√a dx. The integral becomes (1/√a)∫₋∞^∞ e^(−u²) du = √π/√a = √(π/a). Note this is independent of b, since a full-line integral is unaffected by a horizontal shift.
Source: GATE ME 2022 Session 2 (ME-2) (2022)
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