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

Correct Answer: 3

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:

\[ 8(\alpha + \beta) = 3. \]