How do you use Integration by Substitution to find #int(ln(x))^2/xdx#?

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How do you use Integration by Substitution to find #int(ln(x))^2/xdx#?

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Answer 1 · 2014-09-06T20:53:37

By using the substitution #u=lnx#,
#int{(lnx)^2}/xdx={(lnx)^3}/3+C#

Let #u=lnx#.
By taking the derivative with respect to #x#,
#{du}/{dx}=1/x#
By taking the reciprocal,
#{dx}/{du}=x#
By multiplying by #du#,
#dx=xdu#

Now, we can rewrite the integral in terms of #u#,
#int{(lnx)^2}/xdx =intu^2/x xdu=int u^2 du#
by Power Rule,
#=u^3/3+C#
by putting #u=lnx# back in,
#={(lnx)^3}/3+C#

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How do you use Integration by Substitution to find #int(ln(x))^2/xdx#?

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