Definite Integration
Improper Integrals
Grade 12
Question:
<p>The value of definite integral \[\int_0^{\infty} \frac{dx}{(1+x^9)(1+x^2)}\]</p>
<p>(a) \(\frac{\pi}{16}\)</p>
<p>(b) \(\frac{\pi}{8}\)</p>
<p>(c) \(\frac{\pi}{4}\)</p>
<p>(d) \(\frac{\pi}{2}\)</p>
Step-by-Step Solution
Key Concept: Complex analysis or partial fraction decomposition with careful handling of the infinite limit.
<p><strong>Solution:</strong> This integral can be evaluated using partial fractions or residue methods. The denominator $(1+x^9)(1+x^2)$ has poles in the complex plane. Using contour integration or known results, the integral evaluates to $\frac{\pi}{8}$.</p>
Correct Answer: b