Indefinite Integration
Integration by Substitution
Grade 12
Question:
<p>Let <span class="math">\(f(x) = \int \frac{x^2 \, dx}{(1+x^2)(1+\sqrt{1+x^2})}\)</span> and <span class="math">\(f(0) = 0\)</span>, then the value of <span class="math">\(f(1)\)</span> is</p>
<p>(a) <span class="math">\(\log(1+\sqrt{2})\)</span></p>
<p>(b) <span class="math">\(\log(1+\sqrt{2}) - \frac{\pi}{4}\)</span></p>
<p>(c) <span class="math">\(\log(1+\sqrt{2}) + \frac{\pi}{4}\)</span></p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Use substitution to simplify the integrand and apply properties of logarithms.
<p><strong>Solution:</strong> This requires integration by substitution and logarithmic properties.</p>
Correct Answer: A