If f(a + b - x) = f(x) , then _a^b x f(x) dx is equal to: - Mathematics MCQ AIEEE-CBSE-ENG-03 2026 | ExamTest.live

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If f (a + b - x) = f (x), then \int_{a}^{b} x f (x) d x is equal to:

Solution & Step-by-step Explanation

Let I = \int_{a}^{b} x f (x) d x .Using the property \int_{a}^{b} g (x) d x = \int_{a}^{b} g (a + b - x) d x : I = \int_{a}^{b} (a + b - x) f (a + b - x) d x Given f (a + b - x) = f (x) : I = \int_{a}^{b} (a + b - x) f (x) d x I = (a + b) \int_{a}^{b} f (x) d x - \int_{a}^{b} x f (x) d x I = (a + b) \int_{a}^{b} f (x) d x - I 2 I = (a + b) \int_{a}^{b} f (x) d x ⟹ I = \frac{a + b}{2} \int_{a}^{b} f (x) d x

Try it yourself before checking the explanation above.

If f (a + b - x) = f (x), then \int_{a}^{b} x f (x) d x is equal to:

\frac{a - b}{2} \int_{a}^{b} f (b - x) d x

\frac{a + b}{2} \int_{a}^{b} f (x) d x

\frac{b - a}{2} \int_{a}^{b} f (x) d x

\frac{a + b}{2} \int_{a}^{b} f (a + b - x) d x

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