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Evaluate : int (1)/(4+5 sin x)dx...

Evaluate : `int (1)/(4+5 sin x)dx`

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`int (1)/(4+5 sin x)dx`
` =int (1)/(4(sin^(2).(x)/(2)+cos^(2).(x)/(2))+5.2sin.(x)/(2)cos.(x)/(2))dx`
`=1/2 int(sec^(2).(x)/(2))/(2tan^(2).(x)/(2)+2+5tan.(x)/(2))dx`
(Divide `N^(r) " and " D^(r) " by "cos^(2).(x)/(2))`
`=int(dt)/(2t^(2)+5t +2) " ""Let tan"(x)/(2)=t`
` =int(1)/((t+2)(2t+1)) dt rArr 1/2 sec^(2) .(x)/(2) dx=dt`
`=2/3 int (1)/(2t+t)dt-1/3 int(1)/(t+2)dt`
(By partial fraction)
`=1/3 log |2t+1|-1/3 log|t-2|+c` lt brgt `=1/3 log|(2t+1)/(t+2)| +c -1/3 log|(2tan.(x)/(2) +1)/(tan.(x)/(2)+2)| +c.`
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