Indefinite Integration
Integration of irrational functions
Grade 12
Question:
<p>Evaluate \(\int \frac{x+2}{\sqrt{x^2+2x+3}} dx\)</p>
<p>(a) \(\sqrt{x^2+2x+3} + \log|x+1+\sqrt{x^2+2x+3}| + C\)</p>
<p>(b) \(\sqrt{x^2+2x+3} - \log|x+1+\sqrt{x^2+2x+3}| + C\)</p>
<p>(c) \(\sqrt{x^2+2x+3} + \log|x+1-\sqrt{x^2+2x+3}| + C\)</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: Decompose the numerator as a linear combination of the derivative of the expression under the square root and a constant, then separate the integral into two manageable parts.
<p><strong>Step 1:</strong> Let $x + 2 = A \frac{d}{dx}(x^2+2x+3) + B$</p><p><strong>Step 2:</strong> Then $x + 2 = A(2x+2) + B$</p><p><strong>Step 3:</strong> Equating coefficients: $A = \frac{1}{2}$ and $B = 1$</p><p><strong>Step 4:</strong> Apply the decomposition method and integrate the rational and irrational parts separately.</p><p>∴ Answer is (a) $\sqrt{x^2+2x+3} + \log|x+1+\sqrt{x^2+2x+3}| + C$</p>
Correct Answer: a