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

Answer 1

Christopher Allers

#I = intln(x)dx#

Let #ln(x) = t#

#=> x=e^t#

#=> dx = e^tdt#

Substituting in the Integral,

#I = intte^tdt#

On integrating by parts, keeping the first function as #t# and second function as #e^t#, we get

#I = tinte^tdt - int(dt/dtinte^tdt)dt#

Which is, simply,

#I = te^t - e^t + C#

#=> I = e^t ( t-1) + C#

Substituting the value of #t = ln(x)#,

#intln(x)dx = x[ln(x) - 1] + C#

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Answer 2

Emilia Aguilar

The antiderivative of ( \ln(x) ) is ( x\ln(x) - x + 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.