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What is the antiderivative of #ln(x)^2#?

Answer 1

Isabella Ackerman

#int (lnx)^2dx = x(ln^2x -2lnx + 2 ) + C #

Integrate by parts, using #dx# as differential part:

#int (lnx)^2dx = xln^2x -int x d(ln^2x) #

#int (lnx)^2dx = xln^2x - 2int x lnx/xdx #

#int (lnx)^2dx = xln^2x - 2int lnxdx #

Integrating by parts again:

#int (lnx)^2dx = xln^2x -2 xlnx + 2 int x d(lnx) #

#int (lnx)^2dx = xln^2x -2 xlnx + 2 int x dx/x #

#int (lnx)^2dx = xln^2x -2 xlnx + 2 int dx #

#int (lnx)^2dx = xln^2x -2 xlnx + 2 x + C #

#int (lnx)^2dx = x(ln^2x -2lnx + 2 ) + C #

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Answer 2

Emilia Aguilar

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

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Answer 3

Emilia Aguilar

The antiderivative of (\ln(x)^2) is:

(\frac{1}{3}\ln(x)^3 + 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.