Indefinite Integration
Integration of Rational Functions
Grade 12
Question:
<p>If \(f'(x) + g'(x) = (f(x) + g(x))^2 + 1\), then adding two given equations gives \(\displaystyle\int \dfrac{f'(x)+g'(x)}{(f(x)+g(x))^2+1}\,dx = \displaystyle\int 1\,dx\). What is the result?</p>
<p>\(\sin^{-1}(f(x)+g(x)) = x + C\)</p>
<p>\(\cos^{-1}(f(x)+g(x)) = x + C\)</p>
<p>\(\tan^{-1}(f(x)+g(x)) = x + C\)</p>
<p>\(\cot^{-1}(f(x)+g(x)) = x + C\)</p>
Step-by-Step Solution
Key Concept: Recognize that the integral on the left is a standard arctangent form: ∫du/(u²+1) = arctan(u). Substituting u = f(x) + g(x) directly evaluates the left side, while the right side simply gives x + C.
<p><strong>Step 1:</strong> Recognize the left side has the form ∫du/(u²+1) where u = f(x) + g(x).</p><p><strong>Step 2:</strong> The numerator is d/dx[f(x) + g(x)] = f'(x) + g'(x), confirming the substitution structure.</p><p><strong>Step 3:</strong> Apply the standard formula: ∫du/(u²+1) = arctan(u) + C</p><p><strong>Step 4:</strong> Therefore: arctan(f(x) + g(x)) = x + C</p><p><strong>Step 5:</strong> The result is <strong>arctan(f(x) + g(x)) = x + C</strong></p>
Correct Answer: C