AP EAMCET · Maths · Indefinite Integration
\(\int(\log x)^3 x^4 d x=\)
- A \(x^5\left[\frac{1}{5}(\log x)^3-\frac{3}{25}(\log x)^2+\frac{6}{125} \log x-\frac{6}{625}\right]+c\)
- B \(x^5\left[\frac{1}{5}(\log x)^3-\frac{2}{25}(\log x)^2+\frac{6}{125} \log x-\frac{12}{125}\right]+c\)
- C \(x^5\left[\frac{1}{5}(\log x)^3-\frac{4}{25}(\log x)^2-\frac{9}{125} \log x-\frac{8}{125}\right]+c\)
- D \(x^5\left[\frac{1}{5}(\log x)^3+\frac{3}{25}(\log x)^2-\frac{6}{125} \log x-\frac{6}{125}\right]+c\)
Answer & Solution
Correct Answer
(A) \(x^5\left[\frac{1}{5}(\log x)^3-\frac{3}{25}(\log x)^2+\frac{6}{125} \log x-\frac{6}{625}\right]+c\)
Step-by-step Solution
Detailed explanation
\(\int(\log x)^3 x^4 d x = \frac{x^5}{5}(\log x)^3 - \int \frac{x^5}{5} \cdot 3(\log x)^2 \frac{1}{x} d x\) \(= \frac{x^5}{5}(\log x)^3 - \frac{3}{5} \left[ \frac{x^5}{5}(\log x)^2 - \int \frac{x^5}{5} \cdot 2(\log x) \frac{1}{x} d x \right]\)…
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