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\( \int \frac{x^{n}}{1+\frac{x}{1 !}+\frac{x^{2}}{2 !}+\ldots+\frac{x^{4}}{n !}+\ldots} d x \) |
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Answer» ∵ 1 + \(\frac x{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+....+\frac{x^n}{n!}+...\) = ex ∴ \(\int\frac{x^n}{e^x}dx=\int x^ne^{-x}dx\) = xn ∫e-xdx - ∫(nxn-1(-e-x))dx (∵∫e-x = -e-x) = -xne-x + ∫nxn-1e-xdx = -xn e-x - nxn-1e-x + ∫n(n - 1) xn-2 e-x dx = -xn e-x - nxn-1e-x - n(n-1)xn-2e-x-....-n! e-x = -(xn + nxn-1 + n(n-1)xn-2+.....+n!)e-x |
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