Find the following indefinite integrals, identifying any general rules of calculus that you see, 1) int (x^2/3 ) ln(5x) dx

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to integrate by parts , hence you need to use the formula:

int udv = uv - int vdu

You need to select u as `ln 5x`  and dv as `x^(2/3) dx`  such that:

`u = ln 5x =gt du = (dx)/x`

`dv = x^(2/3) dx =gt v = (x^(2/3 + 1))/(2/3 + 1)`

`v = 3(x^(5/3))/5`

`int (x^(2/3) ) ln 5x dx = (3ln 5x *x^(5/3))/5 - int 3(x^(5/3))(dx)/(5x)`

`int (x^(2/3) ) ln 5x dx = (3ln 5x *x^(5/3))/5 - (3/5) int x^(5/3 - 1)dx`

`int (x^(2/3) ) ln 5x dx = (3ln 5x *x^(5/3))/5 - (3/5) int x^(2/3) dx`

`int (x^(2/3 )) ln 5x dx = (3ln 5x *x^(5/3))/5 - (9/25) x^(5/3) + c`

You need to factor out`(3/5) x^(5/3)`  such that:

`int (x^(2/3)) ln 5x dx = (3/5) x^(5/3)( ln 5x - 3/5) + c`

Hence, evaluating the integral using parts yields `int (x^(2/3)) ln 5x dx = (3/5) x^(5/3)( ln 5x - 3/5) + c.`

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