Integral of Ln xDate: 3/31/96 at 13:28:34 From: Anonymous Subject: Calculus Dr. Math, Can you please tell me what the integral of the natural log of x is (ln x)? Date: 3/31/96 at 22:27:25 From: Doctor Sebastien Subject: Re: Calculus Hello, Here's the question: Integral (ln x) dx Let's substitute ln x by y: x = e^y dx/dy = e^y So Integral (ln x) dx = Integral (ln x) dx/dy dy = Integral (y e^y) dy Using integration by parts, Integral (y e^y) dy = uv - Integral (v du/dy) dy Let u = y. du/dy = 1 Integral dv = Integral (e^y) dy v = e^y Therefore, Integral (y e^y) dy = y e^y - Integral (e^y * 1) dy = y e^y - (e^y) + c, where c is a constant. Therefore, Integral (ln x) dx = y e^y - (e^y) + c = (ln x)e^(ln x) - e^(ln x) + c = x(ln x) - x + c Sometimes you will do integrations of simple terms that don't have a solution that take only one line. You will then have to use integration by parts. -Doctor Sebastien, The Math Forum |
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