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Evaluate: $\int\limits_{0}^{1} x e^{-2x} dx$

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1 Answer

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Toolbox:
  • Bernoulle's formula: $\int udv= uv -u'v_1+u''v_2-v'''v_3+....+(-1)^n u^{(n)}v_n.....$
  • Where $u',u''.....u^{(n)}....$ are successive derivatives of u and $v_1,v_2....$ are successive integrals of v.
$\int\limits_{0}^{1} x e^{-2x} dx$
Step 1:
using Bernoulles formula
$\int\limits_{0}^{1} x e^{-2x} dx$
$u=x\;dv=e^{-2x}dx$
$u'=1 \;v=\large\frac{-e^{-2x}}{2}$
$v_1=\large\frac{e^{-2x}}{4}$
Step 2:
$\qquad= uv-u'v_1\bigg]_0^1$
$\qquad= \large\frac{-xe^{-2x}}{2} -\frac{e^{-2x}}{4} \bigg]_0^1$
Step 3:
$\large\frac{1}{4}-\frac{e^{-2}}{2}-\large\frac{e^{-2}}{4}=\large\frac{1}{4}-\large\frac{3}{4}e^{-2}$
answered Aug 14, 2013 by meena.p
 
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