Indefinite Integration
Integration of cosec^5 x — IBP
nta_pyq_2024_apr
Grade 12
Question:
If $\int \cosec^5 x\,dx = \alpha\cot x\cosec x\!\left(\cosec^2 x+\dfrac{3}{2}\right)+\beta\log_e\!\left|\tan\dfrac{x}{2}\right|+C$, where $\alpha,\beta\in\mathbb{R}$ and $C$ is the constant of integration, then the value of $8(\alpha+\beta)$ equals
Step-by-Step Solution
Key Concept: Use integration by parts on $\int\cosec^3 x\cdot\cosec^2 x\,dx$. Reduce to $\int\cosec^3 x\,dx$ via the standard IBP result, then collect terms.
$\alpha=-\frac{1}{4}$, $\beta=\frac{3}{8}$. $8(\alpha+\beta)=1$.
Correct Answer: 1