Indefinite Integration
Composite Functions
Grade 12
Question:
<p>If \(f(x) = \sqrt{\frac{x + 2}{2x + 3}}\), then evaluate \(\int \frac{f(x)}{x^{1/2}} dx\)</p>
<p>(a) \(-\frac{1}{3} \sin^{-1}\sqrt{\frac{1 + 2f(x)}{2}} + C\)</p>
<p>(b) \(h\left\{\frac{3f(x) + 2}{3f(x) - 2}\right\} + C\)</p>
<p>(c) \(g\left\{\frac{1 + 2f(x)}{1 - 2f(x)}\right\} + C\)</p>
<p>(d) \(g(x) = \tan^{-1}x, h(x) = \log|x|\)</p>
Step-by-Step Solution
Key Concept: Use substitution to convert the composite function into a standard form that integrates to an inverse trigonometric function.
<p>Substitute $f(x) = \sqrt{\frac{x + 2}{2x + 3}}$ and use appropriate trigonometric or inverse trigonometric substitutions. The integral evaluates to $g\left\{\frac{1 + 2f(x)}{1 - 2f(x)}\right\} + C$ where $g(x) = \tan^{-1}x$.</p>
Correct Answer: C