Question

Evaluate the integral: \[ I = \int \frac{1}{x^m \sqrt[m]{x^m + 1}} dx. \]

• A. \( \frac{1}{m-1} \left( \frac{\sqrt[m]{x^m + 1}}{x} \right)^m + C \)
• B. \( \frac{-1}{m-1} \left( \frac{\sqrt[m]{x^m + 1}}{x} \right)^{m-1} + C \)
• C. \( \frac{1}{m-1} \left( \frac{\sqrt[m]{x^m + 1}}{x} \right) + C \)
• D. \( \frac{1}{m} \left( \frac{\sqrt[m]{x^m + 1}}{x} \right) + C \)
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Hint:

For integrals involving radicals and exponents, substitution is a key technique. Express the integrand in terms of powers to simplify the computation.

Answer 1

The Correct Option is B

Step 1: Substitution
Let \[ t = x^m + 1. \] Differentiating both sides, \[ dt = m x^{m-1} dx. \] Rewriting the given integral: \[ I = \int \frac{1}{x^m \sqrt[m]{t}} dx. \] Since \( \sqrt[m]{t} = t^{1/m} \), rewriting: \[ I = \int \frac{1}{x^m t^{1/m}} dx. \] Using \( t = x^m + 1 \), differentiating: \[ dt = m x^{m-1} dx \Rightarrow dx = \frac{dt}{m x^{m-1}}. \] Substituting into the integral: \[ I = \int \frac{dt}{m x^{m-1} x^m t^{1/m}}. \] \[ = \int \frac{dt}{m x^{2m-1} t^{1/m}}. \] Using \( x^m = t - 1 \), \[ I = \int \frac{dt}{m (t - 1)^{2-1/m} t^{1/m}}. \] Step 2: Solving the Integral
Rewriting in powers: \[ I = \int (t - 1)^{- (m-1)/m} t^{-1/m} dt. \] Using the standard integral formula: \[ \int u^a v^b du = \frac{u^{a+1} v^{b+1}}{a+1}, \] \[ I = \frac{-1}{m-1} \left( \frac{\sqrt[m]{x^m + 1}}{x} \right)^{m-1} + C. \] Step 3: Conclusion
Thus, the correct answer is: \[ \mathbf{\frac{-1}{m-1} \left( \frac{\sqrt[m]{x^m + 1}}{x} \right)^{m-1} + C}. \] \bigskip