Evaluate the integral. int x^2 log(4x) dx

Adrianna Macias 2022-07-19 Answered
Evaluate the integral. x 2 log ( 4 x ) d x
The problem is x 2 log ( 4 x ) d x
Here ln refers to the natural logarithm.
So far, I know u = x 2 and d u = 2 x ( d x )
So d v = ln ( 4 x ) d x and v = 1 / x, but I don't know where to go from here.
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Answers (2)

eishale2n
Answered 2022-07-20 Author has 15 answers
Integrating by parts,
x 2 log 4 x d x = 1 3 x 3 log 4 x 1 3 x 3 4 4 x d x = 1 3 x 3 log 4 x 1 3 x 2 d x = 1 3 x 3 log 4 x 1 9 x 3 + C

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Levi Rasmussen
Answered 2022-07-21 Author has 6 answers
x 2 ln ( 4 x ) d x = 1 3 ln ( 4 x ) d ( x 3 ) = 1 3 ( x 3 ln ( 4 x ) x 3 d ( ln ( 4 x ) ) )
x 3 d ( ln ( 4 x ) ) = 4 x 3 4 x d x
Thus simplifying your problem considerably.

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