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If f(a+b-x)=f(x), then prove that...

If `f(a+b-x)=f(x),` then prove that `int_a^b xf(x)dx=(a+b)/2int_a^bf(x)dxdot`

Text Solution

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Let `I=int_{a}^{b} x f(x) d x`
` therefore I=int_{a}^{b}(a+b-x) f(a+b-x) d x,(because int_{a}^{b} f(x) d x=int_{a}^{b} f(a+b-x) d x) `
` Rightarrow I=int_{a}^{b}(a+b-x) f(x) d x`
` Rightarrow I=(a+b) int_{a}^{b} f(x) d x-I[U s i n g(1)] `
` Rightarrow I+I=(a+b) int_{a}^{b} f(x) d x`
` Rightarrow 2 I=(a+b) int_{a}^{b} f(x) d x`
` Rightarrow I=(frac{a+b}{2}) int_{a}^{b} f(x) d x`
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