INT 1/sqrt e (the constant) dy from 1 to infinity.
Determine if it is convergent or divergent
To determine if the integral is convergent or divergent, we need to evaluate it. Let's break it down step by step.
We have the integral ∫(1/√e) dy, with the limits of integration from 1 to infinity. Integrating this expression will give us:
∫(1/√e) dy = (√e)y + C
To evaluate this integral from 1 to infinity, we substitute the upper limit into the antiderivative and subtract the value of the antiderivative at the lower limit:
[(√e)y]1^∞ = (√e)(∞) - (√e)(1)
Now, let's analyze these limits.
- (√e)(∞): As y tends towards infinity, (√e)(y) also tends towards infinity. Therefore, this term diverges to positive infinity.
- (√e)(1): This term is a constant and √e is also a constant. So, this term remains the same.
Since the term (√e)(∞) diverges to positive infinity, the integral as a whole diverges.
Therefore, the given integral ∫(1/√e) dy from 1 to infinity is divergent.