Question

If $\int \cosec^5 x \, dx = \alpha \cot x \cosec x \left( \cosec^2 x + \frac{3}{2} \right) + \beta \log_e \left| \tan \frac{x}{2} \right| + C,$ where $\alpha, \beta \in \mathbb{R}$ and $C$ is the constant of integration, then the value of $8(\alpha + \beta)$ equals:

Answer 1

To evaluate the integral $\int \csc^5 x \, dx$, we use integration by parts. Let

$I = \int \csc^3 x \cdot \csc^2 x \, dx.$

Applying integration by parts, we let:

$I = -\cot x \csc^3 x + \int \cot x \cdot (-3 \csc^2 x \cot x \csc x) \, dx.$

Simplifying, we get:

$I = -\cot x \csc^3 x - 3 \int \csc^3 x (\csc^2 x - 1) \, dx,$ $I = -\cot x \csc^3 x - 3I + 3 \int \csc^3 x \, dx.$

Let

$I_1 = \int \csc^3 x \, dx = -\csc x \cot x - \int \cot^2 x \csc x \, dx.$

Using this and simplifying further, we identify values for $\alpha$ and $\beta$. After solving, we find:

$8(\alpha + \beta) = 3.$