Project ForgeEngineering Mathematics
Complex Variables
GATE 2014
Medium
If f*(x) is the complex conjugate of f(x)=cos(x)+i·sin(x), then for real a and b, ∫ₐᵇ f*(x)f(x)dx is ALWAYS
- Apositive
- Bnegative
- real
- Dimaginary
Explanation
f(x)=e^(ix), so f*(x)f(x)=|f(x)|²=cos²x+sin²x=1 for all x. The integral is simply ∫ₐᵇ1dx=b−a, which is always a real number (it can be positive, negative, or zero depending on a and b, so 'real' is the only ALWAYS-true option).
Source: GATE CH 2014 (2014)
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