Question

Evaluate `int_0^∞ 3^(-4x^2) dx` 

Answer 1

Let l= `int_0^∞ 3^(-4x^2) dx`  

put `3^(-4x^2) =  e^-1` 

taking log on both sides, 

`4x^2log 3 =t`  

`x^2= t / (4 log 3)` 

`x^2 = t/(4 log 3)  =>  x = sqrt t/(2 sqrt( log3))`

diff. w.r.t x, 

`dc= t^(-1/2)/(4sqrt log^3) dt`           `  Lim =[0,∞]>` 

∴ I = `int _0 ^∞ e^-t/(4 sqrt log 3) t^(-1/2)` 

∴` I=1/(4sqrt3)int_0^∞ e^-t. t^(-1/2)dt` 

∴ `I = 1=sqrtpi/(4 log 3)`         ....................`{int _0^∞ e^-t.t^-1/2 dt=sqrtpi}`