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Evaluate: int(sin2x)/(a^2sin^2x+b^2cos^2...

Evaluate: `int(sin2x)/(a^2sin^2x+b^2cos^2x)\ dx`

Text Solution

Verified by Experts

Put `(a^(2)sin^(2)x+b^(2)cos^(2)x)=t` so that
`2(a^(2)-b^(2))sinxcosxdx=dthArrsin2x=(dt)/((a^(2)-b^(2)))`.
`:.I=int(sin2x)/((a^(2)sin^(2)x+b^(2)cos^(2)x))dx=(dt)/((a^(2)-b^(2))t)`
`=(1)/((a^(2)-b^(2)))log|t|+C`
`=(1)/((a^(2)-b^(2)))log(a^(2)sin^(2)x+b^(2)cos^(2)x)+C`.
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