MathB.in
New
Demo
Tutorial
About
your first few lines of work show: \[ \int xe^x\, dx = \sum_{n=2}^k \frac{(-1)^n}{n!}x^ne^x-\int\frac{x^k}{k!}e^x\, dx \] now you take limit as \(k\to\infty\): \begin{eqnarray*} \lim_{k\to\infty}\int xe^x\, dx &=&\lim_{k\to\infty}\left[ \sum_{n=2}^k \frac{(-1)^n}{n!}x^ne^x-\int\frac{x^k}{k!}e^x\, dx\right]\\[10pt] &\overset{?}{=}& \lim_{k\to\infty} \sum_{n=2}^k \frac{(-1)^n}{n!}x^ne^x-\lim_{k\to\infty}\int\frac{x^k}{k!}e^x\, dx\\[10pt] &=& \sum_{n=2}^\infty \frac{(-1)^n}{n!}x^ne^x-\lim_{k\to\infty}\int\frac{x^k}{k!}e^x\, dx\\[10pt] &\overset{??}{=}& \sum_{n=2}^\infty \frac{(-1)^n}{n!}x^ne^x-\int\lim_{k\to\infty}\frac{x^k}{k!}e^x\, dx\\[10pt] &=& \sum_{n=2}^\infty \frac{(-1)^n}{n!}x^ne^x - \int 0\, dx \end{eqnarray*}
ERROR: JavaScript must be enabled to render input!
Mon, 02 Dec 2024 12:56 GMT