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

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

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The correct Answer is:
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`I=int_(a)^(b)xf(x)dx=int_(a)^(b)xf(a+b-x)dx`
( `:'` given `f(x)=f(a+b-x))`
`=int_(a)^(b)(a+b-x)f((a+b)-(a+b-x))dx`
`=int_(a)^(b)(a+b-x)f(x)dx=(a+b)int_(a)^(b)f(x)dx-I`
or `2I=(a+b)int_(a)^(b)int_(a)^(b)f(x) dx` or `I=(a+b)/2 int_(a)^(b)f(x)dx`
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CENGAGE PUBLICATION-DEFINITE INTEGRATION -CAE_TYPE
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