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Evaluate: int(dx)/((1+sinx)^(1/2))...

Evaluate: `int(dx)/((1+sinx)^(1/2))`

Text Solution

Verified by Experts

The correct Answer is:
`sqrt(2) log|(tan ((x)/(4)+(pi)/(x)))|+C`
`int(dx)/((1+sinx)^(1//2))=int(dx)/("cos"(x)/(2)+"sin"(x)/(2))`
`=(1)/(sqrt(2))int(dx)/("sin"((x)/(2)+(pi)/(4)))`
`=(1)/(sqrt(2))int"cosec"((x)/(2)+(pi)/(4))dx`
`=(1)/(sqrt(2))(log|tan((x)/(4)+(pi)/(8))|)/((1)/(2))+C`
`=sqrt(2) log|(tan ((x)/(4)+(pi)/(x)))|+C`
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