TS EAMCET · Maths · Definite Integration
\(\begin{aligned} & \text { If } \int x^4(\log x)^3 d x=x^5\left[A(\log x)^3+B(\log x)^2+C \log x+D\right] \ & +k, \text { then } A+B+C+5 D=\end{aligned}\)
- A \(\frac{2}{25}\)
- B \(\frac{8}{25}\)
- C \(\frac{12}{125}\)
- D \(\frac{16}{125}\)
Answer & Solution
Correct Answer
(A) \(\frac{2}{25}\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \int x_{\text {II }}^4(\log x)^3 d x \\ = & (\log x)^3 \cdot \int x^4 d x-\int\left(3(\log x)^2 \cdot \frac{1}{x} \cdot \int x^4 d x\right) d x \\ = & \frac{x^5}{5}(\log x)^3-\frac{3}{5} \int(\log x)^2 x^4 d x \\ = & \frac{x^5}{5}(\log…
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