Question

If `int (sin2x)/(sin^4x + cos^4x) "d"x = tan^-1["f"(x)] + "c"`, then `"f"(pi/3)` = ______.

Options

• 1

• 2

• 3

• `1/3`

Answer 1

If `int (sin2x)/(sin^4x + cos^4x) "d"x = tan^-1["f"(x)] + "c"`, then `"f"(pi/3)` = 3.

Explanation:

Let I = `int (sin2x)/(sin^4x + cos^4x) "d"x`

= `int (2sinx cosx)/(sin^4x + cos^4x) "d"x`

= `int (2tanxsec^2x)/(1 + tan^4x)x`

Put tan²x = t

⇒ 2 tan x sec²x dx = dt

∴ I = `int "dt"/(1 + "t"^2)`

= `tan^-1 "t" + "c"`

= `tan^-1 (tan^2x) + "c"`

Comparing with `tan^-1["f"(x)] + "c"`, we get f(x) = tan²x

∴ `"f"(pi/3) = tan^2  pi/3`

= `(sqrt(3))^2`

= 3