Question

How do you find the integral of `f(x)=x^(2/3) *ln x` using integration by parts?

Answer 1

`(3x^(5/3))/5lnx-9/25x^(5/3)+C`

Explanation:

Integration by parts formula

`I=intcolor(red)(u)color(blue)((dv)/(dx))dx=color(blue)(u)color(red)(v)-intcolor(blue)(v)color(red)((du)/(dx))dx`

the key to success with this formula is the correct choice of `u" & " (dv)/(dx)`

we are given

`intx^(2/3)lnxdx`

`color(red)(u=lnx=>(du)/(dx)=1/x)`

`color(blue)((dv)/(dx)=x^(2/3)=>v=(3x^(5/3))/5)`

`:.I=color(blue)((3x^(5/3))/5)color(red)(lnx)-intcolor(blue)((3x^(5/3))/5)xxcolor(red)(1/x)dx`

`:.I=(3x^(5/3))/5lnx-3/5int(x^(2/3))xdx`

`:.I=(3x^(5/3))/5lnx-3/5xx3/5x^(5/3)+C`

`:.I=(3x^(5/3))/5lnx-9/25x^(5/3)+C`