Question

Integrate the following with respect to x.

xⁿ log x

Answer 1

`int x^"n" log x  "d"x = int  "udv"`

`int "udv" = "uv" - int "vdu"`

`int x^"n" log x  "d"x = (log x) [x^("n" + 1)/("n"+ 1)] - int (x^("n" + 1)/("n" + 1)) (("d"x)/x)`

= `(x^("n" + 1)/("n"+ 1)) log x - 1/(("n" + 1)) int x^"n"  "d"x`

= `(x^("n" + 1)/("n" + 1)) log x- 1/(("n" +  1)) (x^("n" + 1)/("n" + 1)) + "c"`

= `(x^("n" + 1)/("n" + 1)) [log x - 1/(("n"+ 1))] + "c"`

Successive derivatives	Repeated Integrals
Take u = log x `"du"/("d"x) = 1/x` du = `("d"x)/x`	dv = `x^"n"  "d"x` `int "dv" = int x^"n"  "d"x` v = `(x^("n" + 1)/("n" + 1))`