AP EAMCET · Maths · Indefinite Integration
Solve \(I_n+n I_{n-1}\), if \(I_n=\int(\ln x)^n d x\)
- A \(x(\ln x)^{n-1}+k\)
- B \(x(\ln x)^n+k\)
- C \(\frac{(\ln x)^n}{x}+k\)
- D \(\frac{(\ln x)^{n-1}}{x}+k\)
Answer & Solution
Correct Answer
(B) \(x(\ln x)^n+k\)
Step-by-step Solution
Detailed explanation
\begin{aligned} I_n & =\int(\log x)^n \\ I_n & =(\log x)^n \int d x-\int\left(\frac{d(\log x)^n}{d x} \int d x\right) d x+k \\ I_n & =x(\log x)^n-\int \frac{n(\log x)^{n-1}}{x} \times x+k \\ I_n & =x(\log x)^n-n I_{n-1}+k \\ \Rightarrow \quad I_n & =n I_{n-1}=x(\log x)^n+k…
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