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Evaluate the following integrals in...

Evaluate the following integrals
`int (dx)/(sin x + sqrt(3) cos x)`

Text Solution

Verified by Experts

Let 1=`rcostheta` and `sqrt(3) = rsintheta rArr r=sqrt((1)^(2)+sqrt(3)^(2))=2`
`tantheta=sqrt(3) rArr theta=pi/3`
`therefore int(dx)(sinx+sqrt(3)cosx) = 1/rint(dx)/(sinxcostheta+cosxsintheta) = 1/rint(dx)/(sin(x+theta))`
`=1/rint"cosec"(x+theta)dx = 1/r ln|tan(x/2+theta/2)|=1/2ln|tan(x/2+pi/6)|+C`
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