integration

Using integration by parts,

∫ ln(5x^2) dx

=xln(5x^2) - ∫ x d[ln(5x+2)]

=xln(5x^2) - ∫ 5x/(5x+2) dx

=xln(5x^2) - ∫ 1 - 2/(5x+2) dx

=xln(5x^2) - [ x - 2(1/5)ln(5x+2)]+C

=(x+2/5)ln(5x+2) - x + C


2010-12-22 13:00:03 補充：
Sorry..Typing mistake in calculation..

∫ ln(5x^2) dx
=xln(5x^2) - ∫ x d[ln(5x^2)]
=xln(5x^2) - ∫ x [10x/5x^2] dx
=xln(5x^2) - ∫ 2 dx
=xln(5x^2) - 2x + C
=x [ln(5x^2)-2] + C

Integrate ln(5x^2) dx
= Integrate ln5 dx + Integrate 2lnx dx
= xln5 + 2 Integrate lnx dx
= xln5 + 2 (xlnx - Integrate x dlnx)
= xln5 + 2xlnx - 2Integrate dx
= xln5 + 2xlnx - 2x + C
= x(ln5 + 2lnx - 2) + C

Alternatively, ln(5x^2) = ln5 + 2lnx
So ∫ln(5x^2)dx = xln5 + 2xlnx - 2x = (ln5 - 2)x + 2xlnx