Math / Calculus

QuestionDetermine whether the following integral is convergent or divergent. If the integral is convergent, find its value. Write the exact answer. Do not round. If it is divergent, write DIV for your answer. $\int_{0}^{\infty} \frac{29 e^{x}}{x} d x$

Studdy Solution

STEP 1

1. The integral in question is an improper integral due to the infinite upper limit.
2. We need to analyze the behavior of the integrand $\frac{29 e^x}{x}$ as $x \to \infty$ .
3. If the integral converges, we will find its exact value; otherwise, we will determine it is divergent.

STEP 2

1. Analyze the behavior of the integrand as $x \to \infty$ .
2. Determine if the integral is convergent or divergent.
3. If convergent, calculate the value of the integral.

STEP 3

Analyze the behavior of the integrand $\frac{29 e^x}{x}$ as $x \to \infty$ .
As $x \to \infty$ , the exponential function $e^x$ grows much faster than the linear function $x$ . Therefore, the integrand $\frac{29 e^x}{x}$ behaves like $e^x$ , which increases without bound.

STEP 4

Determine if the integral is convergent or divergent.
Since the integrand $\frac{29 e^x}{x}$ grows exponentially as $x \to \infty$ , the integral $\int_{0}^{\infty} \frac{29 e^{x}}{x} \, dx$ diverges because the area under the curve becomes infinitely large.
Since the integral is divergent, the answer is $\text{DIV}$ .

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