If ( x)^(-11)/(2)( x)^(-5)/(2) d x= -(p_1)/(q_1)( x)^(9)/(2)-(p_2)/(q_2)( x)^(5)/(2)-(p_3)…

Question

If $\int(\sin x)^{\frac{-11}{2}}(\cos x)^{\frac{-5}{2}} d x= -\frac{p_1}{q_1}(\cot x)^{\frac{9}{2}}-\frac{p_2}{q_2}(\cot x)^{\frac{5}{2}}-\frac{p_3}{q_3}(\cot x)^{\frac{1}{2}}+\frac{p_4}{q_4}(\cot x)^{\frac{-3}{2}}+\mathrm{C}$, where $p_i$ ...

Answer: 16

Accepted Answer

$\begin{gathered} \int (\sin x{)}^{\frac{-11}{2}}(\cos x{)}^{\frac{-5}{2}}dx=\int \frac{1}{{Tan}^{11/2}x}\cdot {\sec }^{8}xdx=\int \frac{{(1+{Tan}^{2}x)}^{3}{\sec }^{2}x}{{Tan}^{11/2}x}dx \ Tanx=t \ \frac{15{p}_{1}{p}_{2}{p}_{3}{p}_{4}}{{q}_{1}{q}_{2}{q}_{3}{q}_{4}}=\frac{15(2	imes 6	imes 6	imes 2)}{9	imes 5	imes 1	imes 3}=16 \end{gathered}$

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