Mathematics

# $\displaystyle\int x^{3}\left ( \log x \right )^{2}dx=\frac{x^{4}}{4}\left ( \log x \right )^{2}-\frac{1}{8}x^{4}\log x+\frac{1}{4k}x^{4}.$ Find the value of $k$

8

##### SOLUTION
$\displaystyle I=\int x^{3}\left ( \log x \right )^{2}dx$

$\displaystyle =\frac{x^{4}}{4}\left ( \log x \right )^{2}-\int2\left ( \log x \right ).\frac{1}{x}.\frac{x^{4}}{4}dx$

$\displaystyle =\frac{x^{4}}{4}\left ( \log x \right )^{2}-\frac{1}{2}\int x^{3}\log dx$

$\displaystyle =\frac{x^{4}}{4}\left ( \log x \right )^{2}-\frac{1}{2}\left [ \frac{x^{4}}{4}\log x-\int \frac{1}{x}\frac{x^{4}}{4}dx \right ]$

$\displaystyle =\frac{x^{4}}{4}\left ( \log x \right )^{2}-\frac{1}{8}x^{4}\log x+\frac{1}{8}\int x^{3}dx$

$\displaystyle =\frac{x^{4}}{4}\left ( \log x \right )^{2}-\frac{1}{8}x^{4}\log x+\frac{1}{32}x^{4}dx$

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One Word Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 109

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