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The integral int(pi//6)^(pi//3)sec^(2//...

The integral `int_(pi//6)^(pi//3)sec^(2//3)x " cosec"^(4//3)x` dx is equal to

A

`3^(5//6)-3^(2//3)`

B

`3^(7//6)-3^(5//6)`

C

`3^(5//3)-3^(1//3)`

D

`3^(4//3)-3^(1//3)`

Text Solution

Verified by Experts

The correct Answer is:
B
Let `I=int_(pi//6)^(pi//3)sec^(2//3)x " cosec"^(4//3) x dx `
`I=int_(pi//6)^(pi//3)(1)/(cos^(2//3)x sin^(4//3)c)dx `
` = int_(pi//6)^(pi//3)(sec^(2)x)/((tanx)^(4//3))dx`
[ multiplying and dividing the denominatior by `cos^(4//3)x]` Put , tan x = t , upper limitt , at `x=pi//3rArrt=sqrt(3)and` lower limit , at `x=pi//6 rArrt=1//sqrt(3)and sec^(2)x dx =` dt
So , `I=int_(1//sqrt(3))^(sqrt(3))(dt)/(t^(4//3))=[(t^(-1//3))/(-1//3)]_(1//sqrt(3))^(sqrt(3))`
`=-((1)/3^(1//6)-3^(1//6))`
`=3*3^(1//6)-3*3^(-1//6)`
`=3^(7//6)-3^(5//6)`
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