How do you evaluate the integral of #(ln x)^2 dx#?

Answer 1
#x(lnx)^2 -2xlnx +2x+C#
To integrate #(lnx)^2#, let #x= e^y# so that #dx= e^y dy#
#int (lnx)^2 dx= int y^2 e^ydy#. Now integrate by parts,
#y^2 e^y -int 2ye^y dy#. Now again integrate by parts,
#y^2 e^y -2[ ye^y- int e^ydy]#
#y^2e^y -2ye^y +2e^y# +C
#x(lnx)^2 -2xlnx +2x+C#
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Answer 2

bp has one great solution Method 1. There are other solutions:

Both of the solution presented below use Integration by Parts. I use the form:

#int u dv = uv-intvdu#.
Both of the solution presented below use #int lnx dx = xlnx - x +C#, which can be done by integration by parts. (And, of course, verified by differentiating the answer.)

Method 2

#int (lnx)^2 dx#
Let #u = (lnx)^2# and #dv = dx#.
Then #du = (2lnx)/x dx# and #v = x#

Integration by parts gives us:

#int (lnx)^2 dx = x(lnx)^2 - 2int lnx dx##
#color(white)"sssssss"# # =x(lnx)^2-2(xlnx - x) +C#
#color(white)"sssssss"# # =x(lnx)^2-2xlnx + 2x +C#

Method 3

#int (lnx)^2 dx = int (lnx)(lnx)dx#
Let #u=lnx# and #dv = lnx dx#
So, #du = 1/x dx# and #v= xlnx -x#

The parts formula gives us:

#int (lnx)^2 dx = (lnx)(xlnx -x)-int(xlnx-x)/x dx#
#color(white)"sssssss"# # =x(lnx)^2-xlnx -int (color(red)(lnx) - color(green)(1))dx#
#color(white)"sssssss"# # =x(lnx)^2-xlnx -(color(red)(xlnx-x) - color(green)(x)) +C#
#color(white)"sssssss"# # =x(lnx)^2-2xlnx +2x +C#
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Answer 3

To evaluate the integral of (ln x)^2 dx, you can use integration by parts. Let u = (ln x)^2 and dv = dx. Then differentiate u to find du and integrate dv to find v. After that, apply the integration by parts formula:

∫ u dv = uv - ∫ v du

Substitute the values of u, v, du, and dv into the formula and solve the integral.

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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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