Question

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

• A. $\frac{x^5}{625}\left[125p^3-75p^2+30p-6\right]+c$ (where, $p=\log x$)
• B. $\frac{x^5}{625}\left[125p^3-25p^2+30p-5\right]+c$ (where, $p=\log x$)
• C. $\frac{x^5}{625}\left[125p^3-60p^2-25p+5\right]+c$ (where, $p=\log x$)
• D. $\frac{x^5}{125}\left[625p^3-75p^2+30p+6\right]+c$ (where, $p=\log x$)

Answer 1

Answer: (A)

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