Indefinite Integration
Integration by Parts
Grade 12
Question:
<p>If \(\int f(x)\,dx = g(x)\), then \(\int f(x^{-1})\,dx\) is equal to</p>
<p>(A) \(g^{-1}(x)\)</p>
<p>(B) \(xf^{-1}(x) - g(f^{-1}(x))\)</p>
<p>(C) \(xf^{-1}(x) - g^{-1}(x)\)</p>
<p>(D) \(f^{-1}(x)\)</p>
Step-by-Step Solution
Key Concept: Apply integration by parts with the inverse function and use the chain rule for derivatives of inverse functions.
<p>Using integration by parts with $u = f^{-1}(x)$ and $dv = dx$, we get $\int f^{-1}(x)\,dx = xf^{-1}(x) - \int x \cdot (f^{-1})'(x)\,dx = xf^{-1}(x) - g(f^{-1}(x))$.</p>
Correct Answer: B