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f,g,h are continuous in [0,a],f(a-x)=f(x...

`f,g,h` are continuous in `[0,a],f(a-x)=f(x),g(a-x)=-g(x),3h(x)-4h(a-x)=5`. Then prove that `int_(0)^(a)f(x)g(x)h(x)dx=0`.

Text Solution

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`I=int_(0)^(a)f(x)g(x)h(x)dx`
`=int_(0)^(a)f(a-x)g(a-x)h(a-x)dx`
`=int_(0)^(oo) f(x)(-g(x))((3h(x)-5)/4)dx`
`=-3/4int_(0)^(a)f(x)g(x)h(x)dx+5/4int_(0)^(a)f(x)g(x)dx`
`=-3/4+5/4int_(0)^(a)f(x)g(x)dx`
`:.I=5/7 int_(0)^(a)f(x)g(x)dx`
`=5/7int_(0)^(a)f(a-x)g(a-x)dx`
`=5/7int_(0)^(a)f(x)(-g)dx`
`=-I`
or `2I=0` or `I=0`
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Knowledge Check

  • If (d)/(dx)f(x)=g(x) , then the value of int_(a)^(b)f(x)g(x)dx is -

    A
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    C
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    D
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  • int [f(x)g"(x) - f"(x)g(x)]dx is

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