What is the antiderivative of #ln(x^3)/x#?

1 Answer
mason m
Dec 3, 2016

#3/2(ln(x))^2+C#

Explanation:

This is the same as asking:

#intln(x^3)/xdx#

Using the logarithm rule #log(a^b)=blog(a)# we can move the constant out. Then, a good substitution would be #u=ln(x)#, which implies that #du=1/xdx# (see this through taking the derivative of #ln(x)#).

#=3intln(x)/xdx=3intln(x)(1/xdx)=3intudu=3u^2/2=3/2(ln(x))^2+C#

We see that integration by parts isn't even necessary!

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