Indefinite Integration
Reduction Formula — sin/cos Fractional Powers
nta_pyq_2026_jan
Grade 12
Question:
If $\displaystyle\int(\sin x)^{-11/2}(\cos x)^{-5/2}\,dx=-\dfrac{p_1}{q_1}(\cot x)^{5/2}-\dfrac{p_2}{q_2}(\cot x)^{-5/2}-\dfrac{p_3}{q_3}(\cot x)^{1/2}+\dfrac{p_4}{q_4}(\cot x)^{-3/2}+C$, where $p_i$ 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 $\dfrac{15p_1p_2p_3p_4}{q_1q_2q_3q_4}$ is equal to
Step-by-Step Solution
Key Concept: Since $m+n=-11/2-5/2=-8$ (negative even), substitute $t=\cot x$. Integral becomes $-\int\tfrac{(1+t^2)^3}{t^{5/2}}dt=-\int(t^{-5/2}+3t^{-1/2}+3t^{3/2}+t^{7/2})dt$.
$\dfrac{15p_1p_2p_3p_4}{q_1q_2q_3q_4}=16$.
Correct Answer: 16