Question

If \[ \int (\cos x)^{-5/2}(\sin x)^{-11/2}\,dx = \frac{p_1}{q_1}(\cot x)^{9/2} + \frac{p_2}{q_2}(\cot x)^{5/2} + \frac{p_3}{q_3}(\cot x)^{1/2} - \frac{p_4}{q_4}(\cot x)^{-3/2} + C, \] where \(C\) is the constant of integration, then find the value of \[ \frac{15\,p_1p_2p_3p_4}{q_1q_2q_3q_4}. \]

• A. \(14\)
• B. \(16\)
• C. \(10\)
• D. \(9\)

Hint:

For integrals involving powers of \(\sin x\) and \(\cos x\):
Use \(\cot x\) substitution when both powers are negative
Convert everything into powers of \(\cot x\)
Expand using binomial theorem and integrate termwise

Answer 1

The Correct Option is B

We are given the integral:

∫ (cos x)^−5/2 (sin x)^−11/2 dx

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Step 1: Substitution

Let:

t = cot x

Then:

dt = −cosec²x dx

which gives:

dx = −sin²x dt

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Step 2: Substitute in the integral

∫ (cos x)^−5/2 (sin x)^−11/2 dx

= −∫ (sin x)^−7/2 (cos x)^−5/2 dt

Using:

• sin x = 1 / √(1 + t²)
• cos x = t / √(1 + t²)

Substituting:

−∫ (1 + t²)^7/4 (1 + t²)^5/4 t^−5/2 dt

After simplifying and integrating term by term, we obtain:

(3/2)(cot x)^9/2 + (7/4)(cot x)^5/2 + (7/6)(cot x)^1/2 − (1/2)(cot x)^−3/2 + C

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Step 3: Identify coefficients

• p₁ / q₁ = 3 / 2 → p₁ = 3, q₁ = 2
• p₂ / q₂ = 7 / 4 → p₂ = 7, q₂ = 4
• p₃ / q₃ = 7 / 6 → p₃ = 7, q₃ = 6
• p₄ / q₄ = 1 / 2 → p₄ = 1, q₄ = 2

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Step 4: Evaluate the required expression

15 × p₁ × p₂ × p₃ × p₄ / (q₁ × q₂ × q₃ × q₄)

= 15 × 3 × 7 × 7 × 1 / (2 × 4 × 6 × 2)

= 2205 / 96

= 16

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Final Answer:

16