Let F ( x ) = ∫ − ∞ x f ( t ) d t ,where x ∈ R , f ≥ 0 is complicated (it cannot be integrated analytically).Can we used the Simpson's rule to approximate this integral, knowing that f ( − ∞ ) = 0?

Question

Let  F  (  x  )  =      ∫          −      ∞        x    f  (  t  )  d  t  ,where   x  ∈      R  ,   f  ≥  0 is complicated (it cannot be integrated analytically).Can we used the Simpson&#039;s rule to approximate this integral, knowing that   f  (  −  ∞  )  =  0?

Oswaldo Rosales · Accepted Answer

Yes. You can do a change of variable in your integral   ξ  =  x  −            1      −      t        t  ,   d  ξ  =      1          t      2        d  t and integrate for   0  &amp;lt;  ξ  &amp;lt;  1.

Damion Hardin · Accepted Answer

Using estimates on   f, you can obtain some   M  &amp;gt;  0 such that       ∫          −      ∞              −      M        f  (  t  )  d  t  &amp;lt;      ε    2   and then use Simpson&#039;s rule with enough points so that you approximate       ∫          −      M        x    f  (  t  )  d  t with an error smaller than       ε    2  . This guarantees an overall error smaller than   ε.

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