For x∈ (-π/2,π/2), if y(x)=∫ (cosecx+sin x)/(cosecx sec x+tan x sin² x)dx, and limy(x)=0 then y(π/4) is equal to

Solution

Maths sir · 4 steps

Write everything in sin and cos, put t = sin x, then u = t - 1/t finishes the job.

1. Rewrite in sin, cos. The numerator is (1+sin² x)/(sin x) and the denominator is 1/(sin x cos x)+(sin³ x)/(cos x) = (1+sin⁴ x)/(sin x cos x). Dividing gives the integrand ((1+sin² x)cos x)/(1+sin⁴ x).
2. Substitute sin x. Put t = sin x, so dt = cos x dx and the integral becomes ∫ (1+t²)/(1+t⁴) dt. Divide top and bottom by t² to get (1+1/t²)/((t-1/t)²+2), and put u = t-1/t, so du = (1+1/t²)dt.

   y = ∫ du/(u²+2) = 1/√2 tan⁻¹((sin x - cosec x)/√2) + C

3. Fix the constant. As x → (π/2)⁻, both sin x and cosec x tend to 1, so the argument tends to 0 and y → C. The given limit is 0, so C = 0.
4. So. At x = π/4, sin x - cosec x = 1/√2 - √2 = -1/√2. Dividing by √2 gives -1/2, so y(π/4) = 1/√2 tan⁻¹(-1/2), which is option (d).

The idea to keep. For (1+t²)/(1+t⁴), divide by t² and put u = t - 1/t. For (1-t²)/(1+t⁴), use u = t + 1/t. Spot this pair on sight.

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