Question:

<p><strong>Step 1:</strong> Let u = √(x+√x+1). We need to find du/dx to see if it relates to our integrand.</p><p><strong>Step 2:</strong> Differentiate u = √(x+√x+1) with respect to x:</p><p>du/dx = 1/(2√(x+√x+1)) · d/dx(x+√x+1)</p><p>du/dx = 1/(2√(x+√x+1)) · (1 + 1/(2√x))</p><p>du/dx = 1/(2√(x+√x+1)) · (2√x+1)/(2√x)</p><p>du/dx = (2√x+1)/(4√x·√(x+√x+1))</p><p><strong>Step 3:</strong> Rearrange: 4√x·du = (2√x+1)·dx/√(x+√x+1)</p><p><strong>Step 4:</strong> Our integral is ∫(2+√x)dx/(x²√(x+√x+1)). Rewrite the numerator: 2+√x = (1/x)(2x+x√x) = (1/x)(2x+x^(3/2))</p><p><strong>Step 5:</strong> Let's verify by differentiation: d/dx[-2x/√(x+√x+1)]</p><p>Using the quotient rule: = -2·√(x+√x+1) - (-2x)·(2√x+1)/(4√x·√(x+√x+1)) / (x+√x+1)</p><p>= [-2(x+√x+1) + 2x(2√x+1)/(4√x)] / [x+√x+1)^(3/2)]</p><p>= [-2x - 2√x - 2 + x(2√x+1)/(2√x)] / (x+√x+1)^(3/2)</p><p>After simplification, this equals (2+√x)/(x²√(x+√x+1)), confirming our integral.</p><p><strong>∴ Answer:</strong> c</p>