If ∫ (sin^(3/2) x+cos^(3/2) x)/√(sin³ x cos³ x sin (x-θ))dx=A√(cos θ tan x-sin θ)+B√(cos θ-sin θ cot x)+C, where C is the integration constant, then AB is equal to

Solution

Maths sir · 4 steps

Split the fraction, then factor sin(x-θ) to match each square root in the answer.

1. Split the integrand. Divide each term of the numerator by the denominator. The first part becomes 1/√(cos³ x sin(x-θ)) and the second becomes 1/√(sin³ x sin(x-θ)).
2. First part gives A. Write sin(x-θ) = cos x(cos θ tan x - sin θ). The first part is then (sec² x)/√(cos θ tan x - sin θ). Put t = cos θ tan x - sin θ, so dt = cos θ sec² x dx, and the integral is 2√t/(cos θ), giving A = 2 sec θ.
3. Second part gives B. In the same way, sin(x-θ) = sin x(cos θ - sin θ cot x), so the second part is (cosec² x)/√(cos θ - sin θ cot x). Put u = cos θ - sin θ cot x, so du = sin θcosec² x dx, and the integral is 2√u/(sin θ), giving B = 2cosecθ.

   AB = 2/(cos θ)×2/(sin θ) = 4/(sin θ cos θ) = 8/(sin 2θ)

4. So. AB = 8cosec(2θ), which is option (b).

The idea to keep. When the answer form is given, read it as a hint: √(cos θ tan x - sin θ) tells you the substitution. Then factor the integrand to match it.

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