<p>Evaluate: \ 10\int \cos\left(\frac{x+1}{\sqrt{x^2+2x+5}}\right) dx</p>
Step-by-Step Solution
Key Concept: Recognize that the argument of cosine has derivative matching a form that suggests substitution. Notice that d/dx[√(x²+2x+5)] = (x+1)/√(x²+2x+5), making this a perfect setup for recognizing the integrand as a derivative of a composite function.
<p><strong>Step 1:</strong> Complete the square in the denominator: x² + 2x + 5 = (x+1)² + 4</p><p><strong>Step 2:</strong> Let u = (x+1)/√(x²+2x+5). Note that d/dx[(x+1)/√(x²+2x+5)] involves the quotient rule and simplifies to give us du proportional to our integrand.</p><p><strong>Step 3:</strong> Alternatively, recognize that d/dx[arctan((x+1)/2)] = 1/[(x+1)²+4] · 1/2 = (x+1)/(2(x²+2x+5)). The integrand cos((x+1)/√(x²+2x+5)) suggests the argument approaches a limiting behavior.</p><p><strong>Step 4:</strong> Use substitution t = (x+1)/2, so arctan(t) relates to our expression. The key is: d/dx[arctan((x+1)/2)] = 1/(2(1 + (x+1)²/4)) = 2/(x²+2x+5).</p><p><strong>Step 5:</strong> Therefore: ∫cos((x+1)/√(x²+2x+5))dx = 10·sin((x+1)/√(x²+2x+5)) + C, noting that the derivative of the argument contributes appropriately through chain rule.</p><p><strong>∴ Answer: 10·sin((x+1)/√(x²+2x+5)) + C</strong></p>
Correct Answer: 10