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

Answer 1

I found: #xln^3(x)-3xln^2(x)+6xln(x)-6x+c#

We can try to evaluate: #int(ln(x))^3dx=# Set #ln(x)=t# #x=e^t# #dx=e^tdt# So we get: #=intt^3e^tdt=# we can try By Parts: #=t^3e^t-int3t^2e^tdt=t^3e^t-3intt^2e^tdt=# Again: #=t^3e^t-3[t^2e^t-int2te^tdt]=# #=t^3e^t-3t^2e^t+6intte^tdt=# Again: #=t^3e^t-3t^2e^t+6te^t-6inte^tdt=# #=t^3e^t-3t^2e^t+6te^t-6e^t+c# But #t=ln(x)# And remember that #e^(ln(x))=x#. So: #=xln^3(x)-3xln^2(x)+6xln(x)-6x+c#

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Answer 2

Scarlett Allison

The antiderivative of (ln(x))^3 is ∫(ln(x))^3 dx = x(ln(x))^3 - 3∫(ln(x))^2 dx + 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.