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Evaluate `int_0^∞ 3^(-4x^2) dx`

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#### Solution

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}`

Concept: Legendre’S Differential Equation

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