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If intf (x)sinx cosx dx=(1)/(2(a^(2)-b^(...

If `intf (x)sinx cosx dx=(1)/(2(a^(2)-b^(2)))log|f(x)|+C`,then f (x)=

A

`(1)/(a^(2)sin^(2)x+b^(2)cos^(2)x)`

B

`(1)/(a^(2)sin^(2)x-b^(2)cos^(2)x)`

C

`(1)/(a^(2)sin^(2)x+b^(2)sin^(2)x)`

D

`(1)/(a^(2)cos^(2)x-b^(2)sin^(2)x)`

Text Solution

Verified by Experts

The correct Answer is:
a
Let `I=int(1)/(a^(2)sin^(2)x+b^(2)cos^(2)x)sinxcosxdx`
`rArrI=(1)/(2(a^(2)-b^(2)))int((a^(2)-b^(2))sin2x)/(a^(2)sin^(2)x+b^(2)cos^(2)x)dx`
`rArr I=(1)/(2(a^(2)-b^(2)))int(1)/(a^(2)sin^(2)x+b^(2)cos^(2)x)d(a^(2)sin^(2)x+b^(2)cos^(2)x)`
`rArr I=(1)/(2(a^(2)-b^(2)))log|a^(2)sin^(2)x+b^(2)cos^(2)x|+C`
Hence , f (x) `=(1)/(a^(2)sin^(2)x+b^(2)cos^(2)x)`
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