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

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Answer 1 · 2016-07-12T19:19:58

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

#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#