Indefinite Integration
Integration leading to Inverse Trigonometric Functions
Grade 12
Question:
<p>Evaluate <span class="math">\(\int \frac{(x+2)\, dx}{(x^2 + 3x + 3)\sqrt{x+1}}\)</span></p>
<p>(a) <span class="math">\(\frac{2}{3} \tan^{-1}\left(\frac{x}{3(x+1)}\right) + C\)</span></p>
<p>(b) <span class="math">\(\frac{2}{\sqrt{3}} \tan^{-1}\left(\frac{x}{\sqrt{3}(x+1)}\right) + C\)</span></p>
<p>(c) <span class="math">\(\frac{1}{3} \tan^{-1}\left(\frac{x}{\sqrt{3}(x+1)}\right) + C\)</span></p>
<p>(d) <span class="math">\(\frac{2}{\sqrt{3}} \tan^{-1}\left(\frac{x}{\sqrt{3}(x+1)}\right) + C\)</span></p>
Step-by-Step Solution
Key Concept: Transform the denominator and use algebraic manipulation followed by substitution to arrive at standard arctangent integral form.
<p>Complete the square in the denominator and use substitution <span class="math">$u = \frac{x}{x+1}$</span> or similar transformation to convert to arctangent form.</p>
Correct Answer: D