How do you integrate #int lnx/sqrtx# by integration by parts method?

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Answer 1 · 2016-07-23T22:03:25

#=2sqrt(x)(ln(x) - 2) + C#

We derive the IBP formula from the product rule:

Take functions #u(x), v(x)#

#(uv)' = u'v + uv'#

#implies uv' = (uv)' - u'v#

#therefore int uv' dx = uv - int u'v dx#

In this case, we have to take #u(x) = ln(x) and v(x) = x^(-1/2)# because we cannot integrate the natural log function directly.

Obtain:

#u'(x) = 1/x and v(x) = 2x^(1/2)#

So #int ln(x)/(sqrt(x))dx = 2x^(1/2)ln(x) - 2int x^(1/2)*1/x dx#

#= 2x^(1/2)ln(x) - 2 int x^(-1/2) dx#

#= 2x^(1/2)ln(x) - 4x^(1/2) + C#

#=2sqrt(x)(ln(x) - 2) + C#