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`

Answer 1

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`