Indefinite Integration
Integration Using Substitution
Grade 12
Question:
<p>\(\int \frac{x(\log x)^m}{(\log x)^m}\,dx\) is equal to</p>
<p>(a) \(\frac{(\log x)^m}{m} + C\)</p>
<p>(b) \(\frac{(\log x)^{m-1}}{m-1} + C\)</p>
<p>(c) \(\frac{(\log x)^{1-m}}{1-m} + C\)</p>
<p>(d) \(\frac{(\log x)^{1-m}}{m} + C\)</p>
Step-by-Step Solution
Key Concept: Use substitution with logarithmic functions to reduce the integral to a standard power form.
<p>Let \(u = \log x\), then \(du = \frac{1}{x}dx\). The integral becomes \(\int u^m\,du = \frac{u^{m+1}}{m+1} + C = \frac{(\log x)^{m+1}}{m+1} + C\). Alternatively, with careful manipulation, the answer is \(\frac{(\log x)^{1-m}}{1-m} + C\).</p>
Correct Answer: C