Question

Decide on what substitution to use, and then evaluate the given integral using a substitution.(Remember to use absolute values where appropriate.)

∫ [(x⁵− x⁴) / (5x6 − 6x5)] dx

Answer 1

The answer is
`(1)/(30)\Big[2\log x-\log 6\Big]`

Explanation:

Given,

`\int (x^(5)-x^(4))/(5x^(6)-6x^5)dx`

`=\int (x^4(x-1))/(x^5(5x-6))dx`

`=(1)/(6)\int\Big[(1)/(x)+(1)/(5x-6)\Big]dx`

`=\fracd{1}{6}\int (1)/(x)dx+(1)/(6* 5)\int(5)/(5x-6)dx`

`=(1)/(30)\Big[5\log x +\log (5x-6)\Big]`

`=(1)/(30)(2\log x-\log 6)`

Hence the result.