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Evaluate: $\int\limits_{0}^{\infty}x^{6}e^{\large\frac{-1}{2}}dx$

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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.
$\int\limits_{0}^{\infty}x^{6}e^{\large\frac{-1}{2}}dx$
Step 1:
Let $t=\large\frac{x}{2}$
$dt=\large\frac{dx}{2} $$=>dx=2dt$
When $x=0\quad t=0$
When $ x = \infty \quad t = \infty$
Step 2:
$\qquad= 2 \int \limits_0^\infty (2t)^6e^{-t} dt$
$\qquad=2^7 \int \limits_0^{\infty} t^6e^{-t}dt$
Step 3:
Using the integral
$\qquad=64 \times \large\frac{6 !}{1^7}$
$\qquad=64 \times 6!$
answered Aug 14, 2013 by meena.p
 
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