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

Options

A. `1/5 ((pi)/(4)-tan ^(-) ((1)/(3 sqrt3)))`
B. `1/10 ((pi)/(4)-tan ^(-) ((1)/(9 sqrt3)))`
C. `pi/40`
D. `1/20 tan ^(-1) ((1)/(9 sqrt3))`

Answer

B

Explanation

`I=int _(pi//6) ^(pi//4) (dx)/(sin 2x (tan ^(5)x + cot ^(5) x ))= int _(pi//6) ^(pi//4) (sec ^(2)x )/(2 tan x (tan ^(5)x + cot ^(5) x ))dx`
Let `tan x =t`
`= int _(1//sqrt3)^(1) (dt)/((t ^(5) + (1)/(t ^(5))))`
`I =1/2int _(1//sqrt3) (t ^(4))/(((t^(5))^(2) +1))dt`
`I = (1)/(10) int _(1//sqrt3) ^(1) (5t^(4))/((t ^(5))^(2) +1)dt`
Let `t ^(5) = u = (1)/(10) int _(1//9 sqrt3)^(1) (du)/(u ^(2) +1)=1/10 tan ^(-1)u |_(1//9 sqrt3) =1/10 ((pi)/(4) -tan ^(-1) (1)/(9 sqrt3))`