The integral ∫1 / 4^(3 / 4) cos (2 cot ⁻¹ √((1-x)/(1+x))) d x is equal to

Solution

Maths sir · 3 steps

Put x = cos 2θ: the root becomes tan θ and the whole integrand turns into -x.

1. Substitute. Let x = cos 2θ with 0 < θ < π/2. Then 1 - x = 2 sin² θ and 1 + x = 2 cos² θ, so √((1-x)/(1+x)) = tan θ. It is positive because θ is in the first quadrant.
2. Simplify inverse. For tan θ > 0, cot⁻¹(tan θ) = π/2 - θ. So 2 cot⁻¹√((1-x)/(1+x)) = π - 2θ, and cos(π - 2θ) = -cos 2θ = -x.

   ∫_(1/4)^(3/4) (-x) dx = -1/2[x²]_(1/4)^(3/4) = -1/2(9/16 - 1/16) = -1/4

3. So. The integrand is just -x, and the integral equals -1/4. That is option (b).

The idea to keep. When you see √((1-x)/(1+x)), put x = cos 2θ. The root becomes tan θ, and the inverse function usually unwraps into something you can integrate in one line.

Answer from the NTA final answer key. Solution by PrepWiser.