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The integral int(pi//6)^(pi//4)dx/(sin2x...

The integral `int_(pi//6)^(pi//4)dx/(sin2x(tan^5x+cot^5x))`equal

A

`(1)/(5)((pi)/(4)-tan^(-1)((1)/(3sqrt(3))))`

B

`(1)/(20)tan^(-1)((1)/(9sqrt(3)))`

C

`(1)/(10)((pi)/(4)-tan^(-1)((1)/(9sqrt(3))))`

D

`(pi)/(40)`

Text Solution

Verified by Experts

The correct Answer is:
C
Let `I=int_(pi//6)^(pi//4)(dx)/(sin2x(tan^(5)x+cot^(5)x))`
`=int_(pi//6)^(pi//4)((1+tan^(2)x)tan^(5)x)/(2 tan x (tan ^(10)x+1))`[`":'sin 2x = (2 tanx)/(1+tan^(2)x)]`
`=(1)/(2)int_(pi//6)^(pi//4)(tan^(4)xsec^(2)x)/((tan ^(10x+1)))dx`
Put `tan^(5)x=t [:'sec^(2)x=1+tan^(2)x]`
`rArr5 tan ^(4)x sec^(2)x dx = dt`
`{:(x,(pi)/(6),(pi)/(4)),(t,((1)/sqrt(3))^(5),1):}`
`:. I=(1)/(2)*(1)/(5)int_((1//sqrt(3))^(5))^(1)(dt)/(t^(2)+1)=(1)/(10)(tan^(-1)(t))_((1sqrt(3))^(5))^(1)`
`=(1)/(10)((pi)/(4)-tan^(-1)((1)/(9sqrt(3))))`
`=(1)/(10)((pi)/(4)-tan^(-1)((1)/(9 sqrt(3))))`
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