If `int_(0)^(x) (bt cos 4 t - a sin 4t)/( t^(2))dt=(a sin 4x)/(x) "for all" x ne0` , then a and b are given by

Options

A. `a=(1)/(4),b=1`
B. `a=2,b=2`
C. `a=-1,b=4`
D. `a=2,b=4`

Answer

A

Explanation

We have,
`underset(0)overset(x)int (bt cos 4 t - a sin 4t)/( t^(2))dt=(a sin 4x)/(x) `
Differentiating both side w.r. to x, we get
` (bt cos 4 t - a sin 4t)/( x^(2))dt=(4ax cos 4x-a sin 4x)/(x^(2)) "for all" x ne0`
`rArr (b-4a) x cos 4 x=0 "for all" x ne 0`
`rArr b-4a=0 rArr b=4a`.
Clearly, `a=(1)/(4) and b=1` satisfy it.