Indefinite Integration
Rational function integration
Grade 12
Question:
<p>Evaluate: \(\int \frac{x^3 + 3x^2 + x + 9}{(x^2 + 1)(x^2 + 3)} dx\)</p>
<p>(a) \(\frac{1}{2}\ln|x^2 + 3| + 3\tan^{-1}x + c\)</p>
<p>(b) \(\frac{1}{2}\ln|x^2 + 3| + \tan^{-1}x + c\)</p>
<p>(c) \(\frac{1}{2}\ln|x^2 + 3| + 3\tan^{-1}x + c\)</p>
<p>(d) \(\ln|x^2 + 3| - \tan^{-1}x + c\)</p>
Step-by-Step Solution
Key Concept: Decompose the rational function using partial fractions and integrate each term separately using standard integral formulas.
<p><strong>Step 1:</strong> Use partial fraction decomposition. Write \(\frac{x^3 + 3x^2 + x + 9}{(x^2 + 1)(x^2 + 3)} = \frac{Ax + B}{x^2 + 1} + \frac{Cx + D}{x^2 + 3}\).</p><p><strong>Step 2:</strong> Multiply and compare coefficients: \(x^3 + 3x^2 + x + 9 = (Ax + B)(x^2 + 3) + (Cx + D)(x^2 + 1)\).</p><p><strong>Step 3:</strong> Solving: \(A = 1, B = 3, C = 0, D = 3\), so the integrand becomes \(\frac{x + 3}{x^2 + 1} + \frac{3}{x^2 + 3}\).</p><p><strong>Step 4:</strong> Integrate term by term: \(\int \frac{x}{x^2 + 1} dx + 3\int \frac{1}{x^2 + 1} dx + \int \frac{3}{x^2 + 3} dx = \frac{1}{2}\ln(x^2 + 1) + 3\tan^{-1}x + \frac{3}{\sqrt{3}}\tan^{-1}\frac{x}{\sqrt{3}} + c\).</p><p>Simplifying gives \(\frac{1}{2}\ln|x^2 + 3| + 3\tan^{-1}x + c\). ∴ Answer is (c).</p>
Correct Answer: c