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Evaluate: int0^pi x/(a^2cos^2x+b^2sin^2x...

Evaluate: `int_0^pi x/(a^2cos^2x+b^2sin^2x)dx`

Text Solution

Verified by Experts

Let I= `int_0^pi x/(a^2cos^2x+b^2sin^2x)dx`
I= `int_0^pi (pi-x)/(a^2cos^2x+b^2sin^2x)dx`
2I=`int_0^pi pi/(a^2cos^2x+b^2sin^2x)dx`
`2I=2πint _0^(π/2)(sec^2x​)/(​a^2+b^2tan^2x)dx`
b tan x = t
`bsec^2xdx=dt`
`I=π/6​int_0^∞ dt/(​a^2+t^2)`
`​I=(pi^2)/(2ab)`
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