PROBLEM $2$

An interesting idea has been given in Iri at all$[1]$ to deal with end point singular,ties in the integral

$I={\displaystyle \underset{a}{\overset{b}{}}}f(x)dx$

Define

$(t)=$exp$(\frac{-c}{1-t^2})$

$(t)=a+\frac{b-a}{}{\displaystyle \underset{-1}{\overset{t}{}}}(u)du$

Where $c$ is a positive constant and

$={\displaystyle \underset{-1}{\overset{1}{}}}{\displaystyle \underset{-1}{\overset{t}{}}}(u)du$

As t varies from $-1$ to $1,(t)$ varies from a to $b.$ Using $x=(t)$ as a change of variable, we obtain

$I={\displaystyle \underset{-1}{\overset{1}{}}}f((t)){}^{\text{'}}(t)$

The function

${}^{\text{'}(t)}=\frac{b-a}{}(t)$

is infinitely differentiable on $-1,1.$

Develop a program code in your preffered language fort he algorithm defined above and solve the

example problem

$I={\displaystyle \underset{0}{\overset{-1}{}}}\sqrt[\phantom{2}]{-ln(x)}dx$

Take $c=4$ for your calculations.

Reference:

Iri. M$.$ S$.$ Moriguti and Y$.$ Takasawa $(1970).$ On numerical integration formula $($in JApanese$)$ J$.$ Res. Inst.

Math. Sci. $91.82.$ Kyoto Univ. Kyoto JApan