Integral Calculation with Trigonometric Powers …

Basics and Substitution

Hard

Integral Calculation with Trigonometric Powers

If $\int {{{(\sin x)}^{\frac{{ - 11}}{2}}}} {(\cos x)^{\frac{{ - 5}}{2}}}dx =$ $- \frac{{{p_1}}}{{{q_1}}}{(\cot x)^{\frac{9}{2}}} - \frac{{{p_2}}}{{{q_2}}}{(\cot x)^{\frac{5}{2}}}-\frac{{{p_3}}}{{{q_3}}}{\left( {\cot x} \right)^{\frac{1}{2}}} + \frac{{{p_4}}}{{{q_4}}}{(\cot x)^{\frac{{ - 3}}{2}}} + C$ , where p_i and q_iare positive integers with gcd(p_i, q_i) = 1 for i = 1, 2, 3, 4 and C is the constant of integration, then $\frac{{15{p_1}{p_2}{p_3}{p_4}}}{{{q_1}{q_2}{q_3}{q_4}}}$ is equal to _______.

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