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Prove the following properties of defini...

Prove the following properties of definite integrals :
`int_(0)^(a)f(x)dx=int_(0)^(a)f(a-x)dx`

Text Solution

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We shall use the following results :
`int_(a)^(b)f(x)dx=int_(a)^(b)f(t)dt" …(1)"`
and `int_(a)^(b)f(x)dx=-int_(b)^(a)f(x)dx" …(2)"`
Consider `int_(0)^(a)f(x)dx`
Put `x=a-t,` Then `dx=-dt`
When `x=0, a-t =0 " "therefore" "t=a.` When `x = a, a-t =a" "therefore t =0`
`therefore" "int_(0)^(a)f(x)dx=int_(a)^(0)f(a-t)(-dt)=-int_(a)^(0)f(a-t)dt`
`=int_(0)^(a)f(a-t)dt`
`=int_(0)^(a)f(a-x)dx`
`therefore int_(0)^(a)f(x)dx=int_(0)^(a)f(a-x)dx`
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Knowledge Check

  • int_(0)^(2a)f(x)dx-int_(0)^(a)f(x)dx=

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

  • If int_(0)^(2a) f(x)dx=int_(0)^(2a) f(x)dx , then

    A
    `f(2a-x)=-f(x)`
    B
    `f(2a-x)=f(x)`
    C
    f(x) is an odd function
    D
    f (x) is an even function

  • If : int_(0)^(2a)f(x)dx=2.int_(0)^(a)f(x)dx , then :

    A
    `f(2a-x)=-f(x)`
    B
    `f(2a-x)=f(x)`
    C
    `f(a-x)=-f(x)`
    D
    `f(a-x)=f(x)`

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