What is int ln4x^2dxln4x2dx?

2 Answers
May 5, 2018

I tried this:

Explanation:

I used some properties of logs first to manipulate the integrand:
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May 5, 2018

int ln(4x^2)\ dx = x [ ln ( 4 x^2 ) - 2 ] + C

Explanation:

The key to solving this integral is to know that

ln(4x^2) = ln 4 + ln x^2 = ln 4 + 2 ln x.

With this, we get

int ln(4x^2)\ dx =
= ln 4 int dx + 2 int ln x\ dx =
= x ln 4 + 2 int ln x\ dx.

Next, we view the remaining integrand as 1 \cdot ln x, and use integration by parts with u' = 1 and v = ln x. With this, we get

x ln 4 + 2 [ x ln x - int dx ] =

= x ln 4 + 2 x ln x - 2x + C =

= x [ ln ( 4 x^2 ) - 2 ] + C.