Step 1: Understanding the Concept:
This problem involves an indefinite integral of an algebraic function involving fractional powers of linear terms.
When we see integrals of the form \( \int \frac{1}{(x-a)^{m/n}(x-b)^{p/q}} dx \), a common substitution strategy is to create a term that looks like a ratio of the two linear factors, say \( t = \frac{x-a}{x-b} \).
In this specific case, the sum of the exponents in the denominator is \( \frac{3}{4} + \frac{5}{4} = \frac{8}{4} = 2 \).
Whenever the sum of these powers equals 2, the substitution \( t = \frac{x-a}{x-b} \) or \( t = \frac{x-b}{x-a} \) will convert the integral into a simple power-rule integration.
Step 2: Key Formula or Approach:
Identify the exponents: \( m = \frac{3}{4} \) and \( n = \frac{5}{4} \).
The integral can be rewritten as:
\[ I = \int \frac{1}{(x - 1)^{3/4} (x + 2)^{5/4}} dx \]
We will manipulate the expression to introduce the ratio \( \frac{x-1}{x+2} \).
Step 3: Detailed Explanation:
1. Rewrite the denominator to group terms:
We can divide and multiply by \( (x+2)^{3/4} \) to create the ratio:
\[ I = \int \frac{1}{(x + 2)^{3/4} \cdot \left(\frac{x - 1}{x + 2}\right)^{3/4} \cdot (x + 2)^{5/4}} dx \]
\[ I = \int \frac{1}{(x + 2)^{3/4 + 5/4} \cdot \left(\frac{x - 1}{x + 2}\right)^{3/4}} dx = \int \frac{1}{(x + 2)^{2} \cdot \left(\frac{x - 1}{x + 2}\right)^{3/4}} dx \]
2. Perform the substitution:
Let \( t = \frac{x - 1}{x + 2} \).
Differentiating both sides with respect to \( x \) using the quotient rule:
\[ \frac{dt}{dx} = \frac{(x + 2)(1) - (x - 1)(1)}{(x + 2)^2} = \frac{x + 2 - x + 1}{(x + 2)^2} = \frac{3}{(x + 2)^2} \]
This gives us \( \frac{dx}{(x + 2)^2} = \frac{dt}{3} \).
3. Substitute back into the integral:
\[ I = \int \frac{1}{t^{3/4}} \cdot \frac{dt}{3} = \frac{1}{3} \int t^{-3/4} dt \]
4. Integrate using the power rule \( \int t^n dt = \frac{t^{n+1}}{n+1} \):
\[ I = \frac{1}{3} \left[ \frac{t^{-3/4 + 1}}{-3/4 + 1} \right] + C = \frac{1}{3} \left[ \frac{t^{1/4}}{1/4} \right] + C \]
\[ I = \frac{1}{3} \cdot 4 \cdot t^{1/4} + C = \frac{4}{3} t^{1/4} + C \]
Step 4: Final Answer:
Substituting back the original expression for \( t \):
\[ I = \frac{4}{3} \left( \frac{x - 1}{x + 2} \right)^{1/4} + C \]
This matches Option (C).