where p i and q i are positive integers with gcd ( p i , q i ) = 1 for i = 1 , 2 , 3 , 4 , and C is the constant of integration, then 15 p 1 p 2 p 3 p 4 q 1 q 2 q 3 q 4 is equal to ______ [2026]

Question

If $\int {(\sin x)}^{\frac{- 11}{2}} {(\cos x)}^{\frac{- 5}{2}} d x = - \frac{p_{1}}{q_{1}} {(\cot x)}^{\frac{9}{2}} -$ $\frac{p_{2}}{q_{2}} {(\cot x)}^{\frac{5}{2}} - \frac{p_{3}}{q_{3}} {(\cot x)}^{\frac{1}{2}} +$ $\frac{p_{4}}{q_{4}} {(\cot x)}^{\frac{- 3}{2}} + C$

where $p_{i}$ and $q_{i}$ are 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 ______ [2026]

Smart Achievers · Accepted Answer

(16)

$\int {(\tan x)}^{- 11 / 2} . \sec^{8} x d x$

$= \int {(\tan x)}^{- 11 / 2} {(1 + \tan^{2} x)}^{3} \sec^{2} x d x$

Put $\tan x = t$

$\Rightarrow \int t^{- 11 / 2} {(1 + t^{2})}^{3} d t$ $= \int t^{- 11 / 2} (1 + 3 t^{2} + 3 t^{4} + t^{6}) d t$

$= \int (t^{- 11 / 2} + 3 t^{- 7 / 2} + 3 t^{- 3 / 2} + t^{1 / 2}) d t$

$= - \frac{2}{9} {(\cot x)}^{9 / 2} - \frac{6}{5} {(\cot x)}^{5 / 2} - 6 {(\cot x)}^{1 / 2} + \frac{2}{3} {(\cot x)}^{- 3 / 2} + C$

$\Rightarrow p_{1} = 2, p_{2} = 6, p_{3} = 6, p_{4} = 2$

$and q_{1} = 9, q_{2} = 5, q_{3} = 1, q_{4} = 3$

$\frac{15 p_{1} p_{2} p_{3} p_{4}}{q_{1} q_{2} q_{3} q_{4}} = \frac{15 \cdot 2 \cdot 6 \cdot 6 \cdot 2}{9 \cdot 5 \cdot 1 \cdot 3} = 16$

Solution

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Solution

Q. If ∫(sinx)-112(cosx)-52dx=-p1q1(cotx)92-p2q2(cotx)52-p3q3(cotx)12+p4q4(cotx)-32+C where pi and qi are positive integers with gcd(pi,qi)=1 for i=1,2,3,4, and C is the constant of integration, then 15p1p2p3p4q1q2q3q4 is equal to ______ [2026]

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