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Question
If f and g are continuous functions in [0, 1] satisfying f(x) = f(a – x) and g(x) + g(a – x) = a, then `int_0^"a" "f"(x) * "g"(x)"d"x` is equal to ______.
Options
`"a"/2`
`"a"/2 int_0^"a" "f"(x)"d"x`
`int_0^"a" "f"(x)"d"x`
`"a" int_0^"a" "f"(x)"d"x`
Solution
If f and g are continuous functions in [0, 1] satisfying f(x) = f(a – x) and g(x) + g(a – x) = a, then `int_0^"a" "f"(x) * "g"(x)"d"x` is equal to `"a"/2 int_0^"a" "f"(x)"d"x`.
Explanation:
Since I = `int_0^"a" "f"(x) * "g"(x)"d"x`
= `int_0^"a" "f"("a" - x) "g"("a" - x)"d"x`
= `int_0^"a" "f"(x)("a" - "g"(x))"d"x`
= `"a" int_0^"a" "f"(x) "d"x - int_0^"a" "f"(x) * "g"(x)"d"x`
= `"a" int_0^"a" "f"(x)"d"x - 1`
or 1 = `"a"/2 int_0^"a" "f"(x)"d"x`
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