Question

How can integrate this? `int_-∞ ^∞(e ^(-4t) ) dt`

Answer 1

Diverges

Explanation:

`int _(-oo) ^(oo) e^(-4t) dt`

`=> lim_(beta to oo) int_(-beta) ^(beta) e^(-4t) dt`

`=> lim_( beta to oo ) [ -1/4 e^(-4t) ]_-beta ^(beta )`

`=> lim_(beta to oo ) { -1/4 e^(-4beta ) - ( -1/4 e^(4beta ) ) }`

`=> lim_(beta to oo ) { 1/4 e^(4beta) - 1/4 e^(-4beta) }`

as `beta -> oo => e^(-4beta ) to 0`

but `beta -> oo => e^(4 beta) to oo`

Hence the integral is divergent

Answer 2

The integral does not converge

Explanation:

`int _(-oo) ^(oo) e^(-4t) dt = int _(-oo) ^0 e^(-4t) dt + int _0 ^(oo) e^(-4t) dt` provided that both integrals converge.

`int _(-oo) ^0 e^(-4t) dt = lim_(ararr-oo) int _a ^0 e^(-4t) dt`

`= lim_(ararr-oo)[-1/4e^(-4t)]_a^0`

`= lim_(ararr-oo)[-1/4 + 1/4e^(-4a)]`

But `lim_(ararr-oo)e^(-4a) = oo`, so the integral does not converge.

Therefore, `int _(-oo) ^(oo) e^(-4t) dt` does not converge.