Function's integralDetermineĀ the integral of the function y=(4+ln x)^3/x

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You may also use the following alternative method, hence, you need to expand the cube, such that:

`(4 + ln x)^3 = 64 + ln^3 x + 12 ln x(4 + ln x)`

`(4 + ln x)^3 = 64 + ln^3 x + 48 ln x + 12 ln^2 x`

`(4 + ln x)^3/x = 64/x + (ln^3 x)/x + (48 ln x)/x + (12 ln^2 x)/x`

`int (4 + ln x)^3/x dx = int (64/x + (ln^3 x)/x + (48 ln x)/x + (12 ln^2 x)/x) dx`

Using thne property of linearity of integral yields:

`int (4 + ln x)^3/x dx = int (64/x) dx + int (ln^3 x)/x dx + int (48 ln x)/x dx + int (12 ln^2 x)/x) dx`

You should come up with the substitution, such that:

`ln x = u => (1/x)dx = du`

Changing the variable yields:

`int (ln^3 x)/x dx = int u^3` du

`int u^3 du = u^4/4 + c`

Substituting back `ln x` for u yields:

`int (ln^3 x)/x dx= (ln^4 x)/4 + c`

`int (12 ln^2 x)/x) dx = 12 (ln^3 x)/3 + c => int (12 ln^2 x)/x) dx = 4(ln^3 x) + c`

`int (48 ln x)/x dx = 48 (ln^2 x)/2 + c => int (48 ln x)/x dx = 24 (ln^2 x) + c`

`int (4 + ln x)^3/x dx = 64 ln |x| + (ln^4 x)/4 + 24 (ln^2 x) + 4(ln^3 x) + c`

Hence, evaluating the given indefinite integral yields `int (4 + ln x)^3/x dx = 64 ln |x| + (ln^4 x)/4 + 24 (ln^2 x) + 4(ln^3 x) + c.`

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