Question

Evaluate: $$ \int \frac{dx}{(x - 3)^{4/5} (x + 1)^{6/5}} = ? $$

• A. \( \frac{5}{4} \sqrt[5]{\frac{x - 3}{x + 1}} + C \)
• B. \( \frac{5}{4} \sqrt[5]{\frac{x + 1}{x - 3}} + C \)
• C. \( \frac{1}{5} \sqrt[5]{\frac{x - 3}{x + 1}} + C \)
• D. \( \frac{5}{4} \sqrt[5]{\frac{x - 3}{x + 4}} + C \)

Hint:

For expressions like \( (x - a)^m (x + b)^n \), try a substitution of \( y = \left( \frac{x - a}{x + b} \right)^r \) to simplify powers.

Answer 1

The Correct Option is A

Let: \[ I = \int \frac{dx}{(x - 3)^{4/5} (x + 1)^{6/5}} \Rightarrow \text{Let } t = \left( \frac{x - 3}{x + 1} \right)^{1/5} \Rightarrow \text{Let } y = \left( \frac{x - 3}{x + 1} \right)^{1/5} \Rightarrow y^5 = \frac{x - 3}{x + 1} \] This is a homogeneous rational function; use substitution: Let: \[ I = \int \left( \frac{1}{(x - 3)^{4/5} (x + 1)^{6/5}} \right) dx \Rightarrow \text{Let } (x - 3)^{1/5} = u,\quad \text{then relate terms} \] Actually, better substitution: Let: \[ u = \left( \frac{x - 3}{x + 1} \right)^{1/5} \Rightarrow u^5 = \frac{x - 3}{x + 1} \Rightarrow x = \frac{3 + u^5 x + u^5}{1 - u^5} \Rightarrow \text{Too complex} \] Instead, try direct integration using known formula: \[ \int \frac{dx}{(x - a)^m (x - b)^n} = \text{if } m + n = 1, \text{ use substitution } \Rightarrow \text{Try differentiating } y = \left( \frac{x - 3}{x + 1} \right)^{1/5} \] Let: \[ y = \left( \frac{x - 3}{x + 1} \right)^{1/5} \Rightarrow \frac{dy}{dx} = \frac{1}{5} \left( \frac{x - 3}{x + 1} \right)^{-4/5} \cdot \frac{(x + 1) - (x - 3)}{(x + 1)^2} = \frac{1}{5} \left( \frac{x - 3}{x + 1} \right)^{-4/5} \cdot \frac{4}{(x + 1)^2} \] Then: \[ dx = \frac{5}{4} (x - 3)^{4/5} (x + 1)^{6/5} dy \Rightarrow \int \frac{dx}{(x - 3)^{4/5} (x + 1)^{6/5}} = \frac{5}{4} \int dy = \frac{5}{4} y + C \] So: \[ \boxed{\int \frac{dx}{(x - 3)^{4/5} (x + 1)^{6/5}} = \frac{5}{4} \left( \frac{x - 3}{x + 1} \right)^{1/5} + C} \]