Question

`int ((x + 1)(x + log x))^4/(3x) "dx" =`______.

Options

• `1/3 (x + log x)^5` + c

• `1/5 (x + log x)^5` + c

• `1/12 (x + log x)^5` + c

• `1/15 (x + log x)^5` + c

Answer 1

`int ((x + 1)(x + log x))^4/(3x) "dx" = underline(1/15 (x + log x)^5 + "c")`.

Explanation:

Put x + log x = t

`=> (1 + 1/x) "dx" = "dt"`

`=> (x +1)/x "dx" = "dt"`

`therefore int ((x + 1)(x + log x))^4/(3x) "dx" = 1/3 int "t"^4`dt

`= 1/3 * "t"^3/5 + "c"`

`= 1/15 "t"^5 + "c"`

`= 1/15 (x + log x)^5 + "c"`