If ( x)^-11/2( x)^-5/2,dx = -p_1q_1( x)^9/2 - p_2q_2( x)^5/2 - p_3q_3( x)^1/2 + p_4q_4( x)…

Question

If $\displaystyle\int(\sin x)^{-11/2}(\cos x)^{-5/2}\,dx = -\dfrac{p_1}{q_1}(\cot x)^{9/2} - \dfrac{p_2}{q_2}(\cot x)^{5/2} - \dfrac{p_3}{q_3}(\cot x)^{1/2} + \dfrac{p_4}{q_4}(\cot x)^{-3/2} + C$, where $p_i,q_i$ are positive integers with $\gcd(p_i,q_i)=1$ for $i=1,2,3,4$, then $\dfrac{15p_1p_2p_3p_4}{q_1q_2q_3q_4}$ is equal to \_\_\_\_.

Answer: 16

Accepted Answer

Write the integrand as $\dfrac{\sec^8x}{\tan^{11/2}x}$ and use $\sec^2x=1+\tan^2x$, so $\sec^8x=(1+\tan^2x)^3\sec^2x$. Substituting $t=\tan x$ ($dt=\sec^2x\,dx$) and expanding $(1+t^2)^3/t^{11/2}$ termwise gives negative powers of $\cot x$ matching the stated form, leading to

$$\frac{15p_1p_2p_3p_4}{q_1q_2q_3q_4} = \frac{15(2\times6\times6\times2)}{9\times5\times1\times3} = 16$$

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