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Prove that int(0)^(oo) (sin^(2)x)/(x^(2)...

Prove that `int_(0)^(oo) (sin^(2)x)/(x^(2))dx=int_(0)^(oo) (sinx)/x dx`

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`I=int_(0)^(oo) "sin"^(2)x . 1/(x^(2))dx`
`=|sin^(2)x(-1/x)|_(0)^(oo) -int_(0)^(oo) 2 sin x cos x (-1/x)dx`
`lim_(x to oo) (sin^(2)x)/x=0` and `lim_(x to oo) (sin^(2)x)/x=lim_(xto oo) (sinx)limt_(x to oo) (sinx)/x=0`
`:.I=0+int_(0)^(oo) (sin2x)/x dx`
`=int_(0)^(oo) (sint)/(t//2) . (dt)/2` (Putting `2x=t`)
`=int_(0)^(oo) (sinx)/x dx`
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CENGAGE-DEFINITE INTEGRATION -JEE Advanced Previous Year
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