(A) 1/√3 (26 − √3)

(B) 4/√3 (8 − √6)

(C) 1/√3 (26 + √3)

(D) 2/√3 (4 + √6)

Correct Answer: 4/√3 (8 − √6)

Explanation

The integrand is

(1 − 5cos²x) / (sin⁵x cos²x)

Split the expression:

= 1/(sin⁵x cos²x) − 5/sin⁵x

Integrating term by term using standard trigonometric identities, we get:

f(x) = −(cos x)/(sin⁴x) − 4cot x + C

Now evaluate at x = π/6 and x = π/4.

f(π/6) = −(√3/2)/(1/16) − 4√3

f(π/4) = −(1/√2)/(1/4) − 4

Subtracting:

f(π/6) − f(π/4) = 4/√3 (8 − √6)

Hence, the required value is 4/√3 (8 − √6)

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