Question:
What is \(\int \frac{\ln x}{x}\,dx\) equal to?
\(\int \frac{\ln x}{x}\,dx\) किसके बराबर है?
Options:
- \(\frac{(\ln x)^2}{2} + c\)
- \(\frac{\ln x}{2} + c\)
- \((\ln x)^2 + c\)
- None of the above
- \(\frac{(\ln x)^2}{2} + c\)
- \(\frac{\ln x}{2} + c\)
- \((\ln x)^2 + c\)
- इनमें से कोई नहीं
Correct Answer: a)
Explanation:
Let \(t = \ln x\). Then \(dt = \frac{1}{x}\,dx\).
The integral becomes:
\[\int t\,dt = \frac{t^2}{2} + c = \frac{(\ln x)^2}{2} + c\]
Hence, the correct option is (a).
माना \(t = \ln x\), तो \(dt = \frac{1}{x}\,dx\)।
समाकल बन जाता है:
\[\int t\,dt = \frac{t^2}{2} + c = \frac{(\ln x)^2}{2} + c\]
अतः सही विकल्प (a) है।