Determine if the improper integral is convergent or divergent, and calculate its value if it is convergent.
$\int_{0}^{\infty} e^{6 x} dx$
Calculate the value of the improper integral. Select the correct choice below and fill in any answer boxes within your choice.
A.
$\int_{0}^{\infty} e^{6 x} dx = ◻ (Type an integer or a fraction. Simplify your answer.)$
B. The integral diverges.

Answer

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Answer

$The integral diverges.$

Steps

Step 1 ：The integral is an improper integral from 0 to infinity of the function $e^{6 x}$ . To determine if it is convergent or divergent, we need to calculate the integral and see if it results in a finite number (convergent) or not (divergent).

Step 2 ：The antiderivative of $e^{6 x}$ is $\frac{1}{6} e^{6 x}$ . We need to evaluate this from 0 to infinity.

Step 3 ：If the limit as x approaches infinity of $\frac{1}{6} e^{6 x} - \frac{1}{6} e^{0}$ is finite, then the integral is convergent. If not, it is divergent.

Step 4 ：The limit as x approaches infinity of $\frac{1}{6} e^{6 x}$ is infinity, which means the integral is divergent.

Step 5 ： $The integral diverges.$

Answer

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