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S_A_V [24]
3 years ago
12

Integrate ln(3x)^2dx

Mathematics
1 answer:
Anestetic [448]3 years ago
5 0
\ln(3x)^2=2\ln(3x)

Take y=3x, so that \dfrac{\mathrm dy}3=\mathrm dx, and you have

\displaystyle\int\ln(3x)^2\,\mathrm dx=\frac23\int\ln y\,\mathrm dy

Integrate by parts, setting u=\ln y and \mathrm dv=\mathrm dy, which give \mathrm du=\dfrac{\mathrm dy}y and v=y. Then

\displaystyle\frac23\int\ln y\,\mathrm dy=\frac23\left(y\ln y-\int\frac yy\,\mathrm dy\right)=\frac23y\ln y-\frac23y+C

which in terms of x is

\displaystyle\frac23y\ln (3x)-\frac23(3x)+C
\displaystyle\frac23y\ln x-2x+C

where the last term uses the property that \ln(ab)=\ln a+\ln b, and the \dfrac23\ln3 term is absorbed into C.
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