Question

Evaluate `int_0^oo e^(-x^2)/sqrtxdx`

Answer 1

Let I = `int_0^oo e^(-x^2)/sqrtxdx`

Put `x^2=t =>x=sqrtt =>sqrtx=t^(1/4)`

Differentiate w.r.t x,

`therefore dx=1/(2sqrtt)dt             lim->[0,oo]`

`therefore "I"=int_0^ooe^(-t)/1t^((-3)/4)/2dt`

`therefore "I"=1/2 int_0^ooe^(-t)t^(1/4 1)dt`

But we know that ,

`int_0^ooe^(-t).t^(n-1)dt`=𝒈𝒂𝒎𝒎𝒂(𝒏)

`therefore "I"=1/2gamma(1/4)`    ……………..{By the definition of gamma fn}