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If f(a+b-x)=f(x), then int(a)^(b)xf(x)dx...

If `f(a+b-x)=f(x)`, then `int_(a)^(b)xf(x)dx` is equal to

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The correct Answer is:
NA
`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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Knowledge Check

  • int_(a)^(b) f(x) dx =

    A
    `int_(a)^(b) f( y ) dy`
    B
    `- int_(a)^(b) f(x) dx `
    C
    `int_(a)^(b) f(a+b-x) dx `
    D
    ` int_(0)^(a) f(a-x) dx `

  • If f(x) is even then int_(-a)^(a)f(x)dx ….

    A
    0
    B
    `2int_(0)^(a)f(x)dx`
    C
    `int_(0)^(a)f(x)dx`
    D
    `-2int_(0)^(a)f(x)dx`

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