TS EAMCET · Maths · Indefinite Integration
\(\int(\log x)^3 x^4 d x\)
- A \(\frac{x^5}{625}\left[125 p^3-75 p^2+30 p-6\right]+c\)
- B \(\frac{x^5}{625}\left[125 p^3-25 p^2+30 p-5\right]+c\) (where, \(p=\log x\) )
- C \(\frac{x^5}{625}\left[125 p^3-60 p^2-25 p+5\right]+c\) (where, \(p=\log x\) )
- D \(\frac{x^5}{125}\left[625 p^3-75 p^2+30 p+6\right]+c\) (where, \(p=\log x\) )
Answer & Solution
Correct Answer
(A) \(\frac{x^5}{625}\left[125 p^3-75 p^2+30 p-6\right]+c\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \text { Let, } I=\int_I^{(\log x)^3 x^4 d x} \\ & =(\log x)^3 \cdot \frac{x^5}{5}-\int \frac{x^5}{5} \cdot 3(\log x)^2 \cdot \frac{1}{x} d x \\ & =\frac{1}{5} x^5(\log x)^3-\frac{3}{5}\left[\int x^4(\log x)^2 d x\right] \\ & =\frac{1}{5} x^5(\log…
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