Questions: Determine whether the integral is convergent or divergent. If it is convergent, evaluate it. (If the quantity diverges, enter DIVERGES.) [ int0^9 frac5sqrt[3]x-1 d x ]

Solution Steps

We need to evaluate the integral \[ \int_{0}^{9} \frac{5}{\sqrt[3]{x-1}} \, dx. \] The integrand \(\frac{5}{\sqrt[3]{x-1}}\) is undefined at \(x = 1\), so we split the integral into two parts: \[ \int_{0}^{1} \frac{5}{\sqrt[3]{x-1}} \, dx \quad \text{and} \quad \int_{1}^{9} \frac{5}{\sqrt[3]{x-1}} \, dx. \]

We evaluate the first part: \[ \int_{0}^{1} \frac{5}{\sqrt[3]{x-1}} \, dx. \] The result of this integral is \(3.75 - 6.49519052838329i\), which indicates that it is not a real number, suggesting divergence.

Next, we evaluate the second part: \[ \int_{1}^{9} \frac{5}{\sqrt[3]{x-1}} \, dx. \] The result of this integral is \(30\), which is a real number and indicates convergence.

Since the first integral diverges (as it yields a complex number), the overall integral \[ \int_{0}^{9} \frac{5}{\sqrt[3]{x-1}} \, dx \] is classified as divergent. Thus, we conclude that the integral diverges.

Final Answer