Question

If \[ \int \frac{1}{\sqrt[5]{(x - 1)^4}(x + 3)^6} \, dx = A \left( \frac{\alpha x - 1}{\beta x + 3} \right)^B + C, \] where \(C\) is the constant of integration, then the value of \(\alpha + \beta + 20AB\) is _______.

Answer 1

Correct Answer: 7

The problem provides the integral: \[ \int \frac{1}{\sqrt[5]{(x - 1)^4 (x + 3)^6}} \, dx = A \left( \frac{\alpha x - 1}{\beta x + 3} \right)^B + C \]

Step 1: Integral Simplification The integral is rewritten as: \[ I = \int \frac{1}{(x - 1)^{4/5} (x + 3)^{6/5}} \, dx \] This is further manipulated to: \[ I = \int \frac{1}{(x - 1)^{4/5} (x + 3)^2} (x + 3)^{4/5} \, dx \]

Step 2: Substitution Application A substitution is defined: \[ t = \frac{x - 1}{x + 3} \] The differential is derived: \[ dt = \frac{4}{(x + 3)^2} \, dx \quad \Rightarrow \quad \frac{dx}{(x + 3)^2} = \frac{dt}{4} \] The integral in terms of 't' becomes: \[ I = \frac{1}{4} \int t^{-4/5} \, dt \]

Step 3: Integration Performed The integral is solved: \[ I = \frac{1}{4} \cdot \frac{t^{1/5}}{1/5} + C \] Substituting back for 't': \[ I = \frac{5}{4} \left( \frac{x - 1}{x + 3} \right)^{1/5} + C \]

Step 4: Comparison with Given Form By comparing the integrated form with the given expression, the constants are identified: \[ A = \frac{5}{4}, \quad \alpha = 1, \quad \beta = 1, \quad B = \frac{1}{5} \]

Step 5: Calculation of the Target Expression The required value is calculated: \[ \alpha + \beta + 20AB = 2 + 20 \times \frac{5}{4} \times \frac{1}{5} \] \[ = 2 + 20 \times \frac{1}{4} = 2 + 5 = 7 \]

Final Result: \[ \boxed{\alpha + \beta + 20AB = 7} \]