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

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

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.

To evaluate the improper integral ∫(e^(-sqrt(t)))/(sqrt(t)) dt from 0 to infinity, we can use a method called the limit of integration. Here's how to do it step-by-step:

Step 1: Split the integral into two parts.
Since the function is not continuous at 0, we need to split the integral into two separate integrals: one from 0 to a positive number, and another from that positive number to infinity.

Step 2: Integrate the first part from 0 to a positive number.
Let's say we integrate from 0 to a positive number b. This gives us:
∫(e^(-sqrt(t)))/(sqrt(t)) dt from 0 to b.

Step 3: Evaluate the integral.
This integral can be evaluated using techniques from calculus. Since the integral is easy for you, we can skip detailed calculations here.

Step 4: Take the limit as b approaches infinity.
After evaluating the integral, we take the limit as b approaches infinity. So we have:
lim(b->∞) ∫(e^(-sqrt(t)))/(sqrt(t)) dt from 0 to b.

Step 5: Simplify the limit.
If the limit converges to a finite value, that will be the evaluation of the entire improper integral. If the limit diverges (i.e., goes to infinity), then the integral is said to be divergent.

In the case of this specific integral, it is necessary to perform further analysis to determine whether it converges or diverges. This requires a more exhaustive evaluation, possibly using a comparison test or other convergence tests.

In summary, the evaluation of this particular improper integral involves splitting it into two parts and then using the limit of integration method to find the final result. Further analysis is needed to determine its convergence or divergence.