Get the answer to "Determine whether the improper integral converges and, if..."

Wierzycaz

Answered question

2021-11-07

Determine whether the improper integral converges and, if so, evaluate it.

\int_{2}^{\infty} e^{- 2 x} d x

Answer & Explanation

Nola Robson

Skilled2021-11-08Added 94 answers

Step 1
The improper integral can be divergent or convergent. There are two methods that can be used to check if the integral is convergent or divergent. In the first method, we use the comparison test with p-integrals, and in the second method, we find the value of the integral.
Step 2
We have the given definite integral as

I = \int_{2}^{\infty} e^{- 2 x} d x

On solving further, we get the result as

\int_{2}^{\infty} e^{- 2 x} d x = {(\frac{- 1}{2} e^{- 2 x})}_{2}^{\infty}

= lim_{x \to \infty} (\frac{- 1}{2} e^{- 2 x}) - lim_{x \to 2} (\frac{- 1}{2} e^{- 2 x})

= \frac{- 1}{2} [(\frac{1}{e^{\infty}}) - (e^{- 2})]

= \frac{- 1}{2} [0 - \frac{1}{e^{2}}]

= \frac{1}{2 e^{2}}

Hence, the given definite integral is convergent, as the value is finite.

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