Simpson’s rule is a numerical method that uses quadratic functions to approximate definite integrals.

With reference to the diagram above, is the area under the cubic curve demarcated by two equal intervals, each of magnitude *h*. The area is estimated using 3 points on the quadratic function , where

Substitute , and in , we have , and . Combining the last two equations, . Substitute this equation and in eq23,

Similarly, if we consider the domain from *h* to *h*‘ (see above diagram),

Therefore, using 5 points on , we have

Extending the logic for *n* points on , we have,

where and .

This implies that there must be an even number of equal intervals for the rule to work, i.e. *n* must be odd.

###### Question

Evaluate

###### Answer

Substituting eq24 in the above equation,

So,

where (form 1)

or equivalently (form 2),

These integrals are useful for computing the numerical solution of the Hartree self-consistent field method for helium.