Question

$\int(\log x)^{3} x^{4} d x$

• A. $\frac{x^{5}}{625}\left[125 p^{3}-75 p^{2}+30 p-6\right]+c$ (where, $p=\log x$)
• 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 1

Answer: (A)

Let, $I=\int(\underset{I}{\log x)^{3}} \underset{II}{x^{4}} d x$
$=(\log x)^{3} \cdot \frac{x^{5}}{5}-\int \frac{x^{5}}{5} \cdot 3(\log x)^{2} \cdot \frac{1}{x} d x$
$=\frac{1}{5} x^{5}(\log x)^{3}-\frac{3}{5}\left[\int x^{4}(\log x)^{2} d x\right]$
$=\frac{1}{5} x^{5}(\log x)^{3}-\frac{3}{5}\left[\frac{x^{5}}{5}(\log x)^{2}-\int \frac{x^{5}}{5} \cdot 2(\log x) \cdot \frac{1}{x} d x\right]$
$=\frac{1}{5} x^{5}(\log x)^{3}-\frac{3}{25} x^{5}(\log x)^{2}+\frac{6}{25} \int x^{4} \log x d x$
$=\frac{1}{5} x^{5}(\log x)^{3}-\frac{3}{25} x^{5}(\log x)^{2}$
$+\frac{6}{25}\left[\frac{x^{5}}{5} \log x-\int \frac{x^{5}}{5} \cdot \frac{1}{x} d x\right]$
$=\frac{1}{5} x^{5}(\log x)^{3}-\frac{3}{25} x^{5}(\log x)^{2}$
$+\frac{6}{125} x^{5} \log x-\frac{6}{625} x^{5}+c$
$=\frac{x^{5}}{625}\left[125 p^{3}-75 p^{2}+30 p-6\right]+c$
where, $p=\log x$