Indefinite Integration
Rational Functions
Grade 12
Question:
<p>Evaluate \(\int \frac{1+x}{\sqrt{1+x^4+x^2}} \cdot \frac{1}{1+x^4} dx\)</p>
<p>(A) \(\ln \frac{x^6}{(x^2-1)} + C\)</p>
<p>(B) \(\frac{1}{6}\ln \frac{x^6}{1+x^{12}} - \frac{1}{6x^6} + C\)</p>
<p>(C) \(\frac{e^x}{3x^{1/3}} - x^{1/6} + \frac{1}{6}x^{-1/2} + C\)</p>
<p>(D) None of these</p>
Step-by-Step Solution
Key Concept: Recognize the structure of the denominator and use algebraic manipulation to separate into integrable parts.
<p>Decompose the integral using partial fractions and substitution techniques to handle the complex denominator.</p>
Correct Answer: B