Indefinite Integration
Integration by substitution
Grade 12
Question:
<p>If \(\int \frac{\log(t+\sqrt{1+t^2})}{\sqrt{1+t^2}}\, dt = \frac{1}{2}(g(t))^2 + C\), where \(C\) is a constant, then \(g(2)\) is equal to</p>
<p>\(2\log(2+\sqrt{5})\)</p>
<p>\(\log(2+\sqrt{5})\)</p>
<p>\(\frac{1}{\sqrt{5}}\log(2+\sqrt{5})\)</p>
<p>\(\frac{1}{2}\log(2+\sqrt{5})\)</p>
Step-by-Step Solution
Key Concept: Recognize that the integrand has the form f'(t)·f(t) where f(t) = log(t+√(1+t²)). The derivative of log(t+√(1+t²)) is exactly 1/√(1+t²), making this a perfect candidate for substitution u = log(t+√(1+t²)), which gives ∫u·du = u²/2.
<p><strong>Step 1:</strong> Let u = log(t+√(1+t²)). Find du/dt:</p><p>du/dt = 1/(t+√(1+t²)) · (1 + t/√(1+t²)) = 1/(t+√(1+t²)) · (√(1+t²)+t)/√(1+t²) = 1/√(1+t²)</p><p><strong>Step 2:</strong> Substitute into the integral:</p><p>∫ [log(t+√(1+t²))]/√(1+t²) dt = ∫ u · du = u²/2 + C = [log(t+√(1+t²))]²/2 + C</p><p><strong>Step 3:</strong> Compare with the given form ½(g(t))² + C:</p><p>(g(t))² = [log(t+√(1+t²))]²</p><p>Therefore: g(t) = log(t+√(1+t²))</p><p><strong>Step 4:</strong> Calculate g(2):</p><p>g(2) = log(2+√(1+4)) = log(2+√5)</p><p>∴ Answer: g(2) = log(2+√5)</p>
Correct Answer: B