can any one explain why this integral diverges ?
integration of [(e^(-sqrt(t)))/sqrt(t)]dt from 0 to infinity
thanks
let u= sqrt(t)
du= 1/2sqrt(t) dt
dt= 2u du
int e^-u * 1/u * 2u du=2 int e^-u du
= -2e^-u eval 0 to inf= 2
check my thinking.
yeah the integration is right
but I was meaning that the integral is improper i.e. the function is not continuous at 0 neither at inf
so how could we evaluate the integration?
thanks for ur reply
To determine why the given integral diverges, we can examine the behavior of the integrand as the variable "t" approaches the infinite limit.
Let's break down the integral:
∫[(e^(-√t))/√t] dt
To assess the divergence, we should consider whether the integrand approaches zero as t tends towards infinity.
As t approaches infinity, √t also approaches infinity. The exponential function e^(-√t) will tend to zero as the argument (√t) becomes infinitely large.
However, the denominator √t remains in the integrand. As t approaches infinity, the denominator also approaches infinity, but at a slower rate than the exponential function approaches zero. Thus, the denominator does not compensate for the exponential decay in the numerator.
Consequently, the integrand does not tend to zero as t goes to infinity. Since the integrand does not converge, the given integral diverges.
In summary, the integral ∫[(e^(-√t))/√t] dt from 0 to infinity diverges because the integrand does not approach zero as the limit of integration tends towards infinity.