Question

f(x) = `int (dx)/(sin^6 x)` is a polynomial of degree

पर्याय

• 5 in cot x

• 5 in tan x

• 3 in tan x

• 3 in cot x

Answer 1

5 in cot x

Explanation:

Let f(x) = `int (dx)/(sin^6x)`

f(x) = `int "cosec"^6x dx`

From the reduction formula, we have

I_n = `int "cosec"^nxdx = -("cosec"^(n - 2)x cot x)/(n - 1) + (n - 2)/(n - 1)I_(n - 2)`

∴ f(x) = `-("cosec"^4xcotx)/5 + 4/5[(-"cosec"^2xcotx)/3 + 2/3I_2]`

= `-("cosec"^4xcotx)/5 - 4/15 "cosec"^2x.cotx + 8/15[-cotx]`

= `(-(1 + cot^2x)^2.cotx)/5 - 4/15(1 + cot^2x)cot x - 8/15(-cot x)`  ...(∵ cosec²x = 1 + cot²x)

= `(-1)/5[1 + cot^4x + 2cot^2x]cot x - 4/15[cot x + cot^3x] - 8/15cotx`

= `(-1)/5[cot x + cot^5x + 2cot^3x] (-4)/15cot x - 4/15cot^3x - 8/15cotx`

= `(-15)/15cot x - (cot^5x)/5 - 10/15cot^3x`

= `(-cot^5x)/5 - 2/3cot^3 x - cotx`

It is a polynomial of degree 5 in cot x.