Compute the following integrals using the substitution method:

a$.{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}x{e}^{x^2}dx$

b$.{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{1}{x\sqrt[\phantom{2}]{ln(x)}}dx$

Calculate the present value of the annuity assuming that it is $(1)$ an ordinary annuity $(2)$ an annuity due. Comparing the two types of annuities, all else equal, which type is more preferable? Why?

$\backslash$table$[[$Amount of annuity,Interest rate,Deposit period $($years$)],[$$$55000,12\%,15]]$

Ordinary, annuity $=374598,$ annuity due $=419549,$ annuity due is better because it discounts for one less year.

Ordinary annuity $=419549,$ annuity due $=374598,$ ordinary annuity is better because it compounds for one more year.

Ordinary annuity $=419549,$ annuity due $=374598,$ ordinary annuity is better because it discounts for one less year.

Ordinary annuity $=374598,$ annuity due $=419549,$ annuity due is better because it compounds for one more year.