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Question
Integrate the following with respect to x.
xn log x
Solution
`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))` |
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