GATE EE 2020 Set 1 — Question 12
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Engineering Mathematics → Calculus → Definite & Improper Integrals
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Question
Which of the following is true for all possible non-zero choices of integers , or all possible non-zero choices of real numbers , as applicable?
Correct answer
(C) (1)/(2π) ∫_(-π)^(π) sin pθ cos qθ dθ = 0
Solution
Consider option (C): The integrand is .
Since is an odd function and is an even function, their product is an odd function because .
The integral of any odd function over a symmetric interval is always zero:This holds for all real non-zero where . Thus, option (C) is always true.
Since is an odd function and is an even function, their product is an odd function because .
The integral of any odd function over a symmetric interval is always zero:This holds for all real non-zero where . Thus, option (C) is always true.
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