Question

How do you find the integral of `sqrt (t)*ln (t) dt`?

Answer 1

use integration by parts to find `intsqrt(t)ln(t)dt=2/3t^(3/2)(ln(t)-2/3)+C`

Explanation:

We use integration by parts by noting that `sqrt(t)=d/dt(2/3t^(3/2))`.
Therefore
`intsqrt(t)ln(t)dt=2/3intln(t)d(t^(3/2))=2/3(ln(t)t^(3/2)-intt^(3/2)dln(t))`.
Since `d/dt(ln(t))=1/t` this gives
`intsqrt(t)ln(t)dt=2/3(ln(t)t^(3/2)-intt^(3/2)/tdt)=2/3(ln(t)t^(3/2)-intt^(1/2)dt)=2/3(ln(t)t^(3/2)-2/3t^(3/2))+C=2/3t^(3/2)(ln(t)-2/3)+C`
With C the integration constant.