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How do you integrate #(x^4)(lnx)#?

Answer 1

Ezra Adams

#intlnx xxx^4dx=x^5/5(lnx-1/5)#

WE can use integration by parts #intudv=uv-intvdu#

Let #u=lnx# and #v=x^5/5#

Hence #du=dx/x# and #dv=x^4dx# and #intudv=uv=intvdu# is

#intlnx xxx^4dx=intudv=uv-intvdu#

= #x^5/5xxlnx-intx^5/5xxdx/x#

= #(lnx xx x^5)/5-1/5intx^4dx#

= #(lnx xx x^5)/5-x^5/25#

= #x^5/5(lnx-1/5)#

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Answer 2

Axel Alvarez

To integrate (x^4 \ln(x)), you can use integration by parts:

[ \int x^4 \ln(x) , dx = \frac{x^5}{5} \ln(x) - \int \frac{x^5}{5} \cdot \frac{1}{x} , dx ]

This simplifies to:

[ \frac{x^5}{5} \ln(x) - \frac{1}{5} \int x^4 , dx ]

Now, integrate (\int x^4 , dx) to get:

[ \frac{x^5}{5} \ln(x) - \frac{x^5}{25} + 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.