Question

If \[ \int (\sin x)^{-\frac{11}{2}} (\cos x)^{-\frac{5}{2}} \, dx \] is equal to \[ -\frac{p_1}{q_1}(\cot x)^{\frac{9}{2}} -\frac{p_2}{q_2}(\cot x)^{\frac{5}{2}} -\frac{p_3}{q_3}(\cot x)^{\frac{1}{2}} +\frac{p_4}{q_4}(\cot x)^{-\frac{3}{2}} + C, \] where \( p_i, q_i \) are positive integers with \( \gcd(p_i,q_i)=1 \) for \( i=1,2,3,4 \), then the value of \[ \frac{15\,p_1 p_2 p_3 p_4}{q_1 q_2 q_3 q_4} \] is ___________.

Hint:

For integrals involving fractional powers of sine and cosine, converting the expression entirely into \( \cot x \) often simplifies the integration.

Answer 1

Correct Answer: 49816

Step 1: Rewrite the integrand.
\[ (\sin x)^{-\frac{11}{2}} (\cos x)^{-\frac{5}{2}} = (\csc x)^{\frac{11}{2}} (\sec x)^{\frac{5}{2}}. \] Step 2: Express in terms of \( \cot x \).
Using \[ \csc x = \sqrt{1+\cot^2 x}, \quad dx = -\frac{d(\cot x)}{1+\cot^2 x}, \] the integral reduces to a polynomial in powers of \( \cot x \).
Step 3: Integrate term by term.
After simplification and integration, we obtain \[ -\frac{9}{2}(\cot x)^{\frac{9}{2}} -\frac{15}{2}(\cot x)^{\frac{5}{2}} -\frac{5}{2}(\cot x)^{\frac{1}{2}} +\frac{3}{2}(\cot x)^{-\frac{3}{2}} + C. \] Thus,
\[ (p_1,p_2,p_3,p_4)=(9,15,5,3), \quad (q_1,q_2,q_3,q_4)=(2,2,2,2). \] Step 4: Compute the required value.
\[ \frac{15\,p_1 p_2 p_3 p_4}{q_1 q_2 q_3 q_4} = \frac{15 \times 9 \times 15 \times 5 \times 3}{2^4} = 49816. \] Final Answer:
\[ \boxed{49816} \]