Definite Integration Question - None

Question

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

Accepted Answer

The integrand is (1/sin x + sin x) / (1/(sin x cos x) + sin x tan² x) = (1+sin² x)/sin x / (1/(sin x cos x) + sin³ x/cos x) = (1+sin² x)/sin x * (sin x cos x) / (1+sin⁴ x) = (cos x (1+sin² x)) / (1+sin⁴ x). Let sin x = t, then cos x dx = dt. The integral becomes ∫ (1+t²) / (1+t⁴) dt = ∫ (1+1/t²) / (t²+1/t²) dt = ∫ d(t-1/t) / ((t-1/t)²+2) = 1/√2 tan^-1((t-1/t)/√2) + C. Using the limit condition, find C and evaluate at x=π/4.

Question:

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