TS EAMCET · Maths · Indefinite Integration
\(I_{m, n}=\int x^m(\log x)^n d x=\)
- A \(\frac{x^{m+1}}{m+1}(\log x)^n-\frac{n}{m+1} I_{m, n-1}\)
- B \(\frac{x^m}{m}(\log x)^n-\frac{n-1}{m+1} l_{m+1, n-1}\)
- C \(\frac{x^{m+1}}{m} \frac{(\log x)^{n+1}}{n+1}-\frac{n}{m+1} I_{m, n-1}\)
- D \(x^m \frac{(\log x)^{n+1}}{n+1}-\frac{n}{m+1} I_{m, n-1}\)
Answer & Solution
Correct Answer
(A) \(\frac{x^{m+1}}{m+1}(\log x)^n-\frac{n}{m+1} I_{m, n-1}\)
Step-by-step Solution
Detailed explanation
Given, \(I_{m, n}=\int x^m(\log x)^n d x\) \(=(\log x)^n \frac{x^{m+1}}{m+1}-\int \frac{x^{m+1}}{m+1} \frac{n(\log x)^{n-1}}{x} d x\) \(\text { (Using integration by parts) }\) \(=\frac{x^{m+1}}{m+1}(\log x)^n-\frac{n}{m+1} \int x^m(\log x)^{n-1} d x\)…
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