<p>Evaluate \(\int \dfrac{2x+3}{\sqrt{1+x+x^2}}\, dx\).</p>
Step-by-Step Solution
Key Concept: Recognize that the numerator 2x+3 can be split into 2(derivative of denominator's argument) plus a constant, enabling substitution and a standard arctrig integral.
<p><strong>Step 1:</strong> Notice that the derivative of (1+x+x²) is (1+2x). Rewrite the numerator:</p><p>2x+3 = (2x+1) + 2 = d/dx(1+x+x²) + 2</p><p><strong>Step 2:</strong> Split the integral:</p><p>∫(2x+3)/√(1+x+x²) dx = ∫(2x+1)/√(1+x+x²) dx + ∫2/√(1+x+x²) dx</p><p><strong>Step 3:</strong> For the first integral, use substitution u = 1+x+x²:</p><p>∫(2x+1)/√(1+x+x²) dx = ∫du/√u = 2√u = 2√(1+x+x²)</p><p><strong>Step 4:</strong> For the second integral, complete the square in the denominator:</p><p>1+x+x² = (x+1/2)² + 3/4</p><p>∫2/√((x+1/2)² + 3/4) dx = 2·sinh⁻¹((x+1/2)/(√3/2)) = 2sinh⁻¹((2x+1)/√3)</p><p>Or equivalently: 2ln|(2x+1)/√3 + √(1+x+x²)| = ln|2x+1+2√(1+x+x²)| - ln(√3)</p><p><strong>Step 5:</strong> Combine both parts:</p><p>∴ Answer: <strong>2√(1+x+x²) + 2sinh⁻¹((2x+1)/√3) + C</strong> or <strong>2√(1+x+x²) + 2ln|(2x+1+2√(1+x+x²))/√3| + C</strong></p>
Correct Answer: 2