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The integral int sec^(2//3) "x cosec"^(4...

The integral `int sec^(2//3) "x cosec"^(4//3)"x dx"` is equal to (here C is a constant of integration)

A

`3(tan x)^(-1//3)+C`

B

`-(tan x)^(-4//3)+C`

C

`-3(cot x)^(-1//3)+C`

D

`-3 (tan x)^(-1//3)+C`

Text Solution

Verified by Experts

The correct Answer is:
D
`int(dx)/((cos x)^(2//3)(sin x)^(4//3))`
`= int ((1)/(cos^(2)x))/(((sin x)/(cos x))^(4//3))dx = int (sec^(2)"x dx")/((tan x)^(4//3))`
Let tan x = t
`sec^(2)x dx = dt`
`int (dt)/(t^(4//3))=int t^(-4//3)dt=(t^(-4//3+1))/(-(4)/(3)+1)`
`=-(3)/(t^(1//3))+c=-(3)/((tan x)^(1//3))+c`
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