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Evaluate the integral of  x^{5}log^{2}x

Option: 1

\frac{x^{6}log^{2}x}{6}+\frac{x^{6}log\ x}{18}+\frac{x^{6}}{108}+C


Option: 2

\frac{x^{6}log^{2}x}{6}-\frac{x^{6}log\ x}{18}+\frac{x^{6}}{108}+C


Option: 3

\frac{x^{6}log^{2}x}{6}-\frac{x^{6}log\ x}{18}-\frac{x^{6}}{108}+C


Option: 4

-\frac{x^{6}log^{2}x}{6}-\frac{x^{6}log\ x}{18}+\frac{x^{6}}{108}+C


Answers (1)

best_answer

Given integral, 
\int x^{5}log^{2}x\ dx
The reduction formula used is,
\int x^{n}log^{m}x\ dx=\frac{x^{n+1}log^{m}x}{n+1}-\frac{m}{n+1}\int x^{n}log^{m-1}x\ dx

\int x^{5}log^{2}x\ dx=\frac{x^{6}log^{2}x}{6}-\frac{2}{6}\int x^{5}log\ x\ dx
\int x^{5}log^{2}x\ dx=\frac{x^{6}log^{2}x}{6}-\frac{1}{3}\left ( \frac{x^{6}log\ x}{6}-\frac{1}{6}\int x^{5}\ dx \right )
\int x^{5}log^{2}x\ dx=\frac{x^{6}log^{2}x}{6}-\frac{x^{6}log\ x}{18}+\frac{x^{6}}{108}+C

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Shailly goel

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