can any one explain how to evaluate this improper integral i.e. the function is not continuous at 0 neither at inf

Question

integration of [(e^(-sqrt(t)))/sqrt(t)]dt from 0 to infinity

the integration part is easy but i want only how to evaluate it

thanks

Answer 1

The indefinite integral is -2 e^(-sqrt(t))

At t = 0, the value of the indefinite integral is -2, and at t = infinity, it approaches zero

Therefore the integral is 0 - (-2) = 2

Integrals to not have to diverge just because the integrand diverges at the endpoints. It all depends upon how fast the functions approach infinity.