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

Answer 1

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

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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Answer 2

The antiderivative of ( \frac{\ln(x^3)}{x} ) with respect to ( x ) is ( \frac{1}{2}(\ln(x))^2 + C ), where ( C ) is the constant of integration.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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