How do you integrate ln(sqrt(x)) ?

1 Answer
Jul 12, 2016

= 1/2 x( ln x - 1) + C

Explanation:

int \ ln(sqrt(x)) \ dx

using IBP: int uv' = uv - int u'v

= int \ ln(sqrt(x)) * (x)' \ dx

= ln(sqrt(x)) * x - int (ln(sqrt(x)))' * x\ dx

= 1/2 x ln x - int \ 1/(sqrt(x)) (1/2 * 1/sqrtx) * x\ dx

= 1/2 x ln x - 1/2 int \ dx

= 1/2 x ln x - 1/2 x + C

= 1/2 x( ln x - 1) + C