Questions: ∫ (3+1 / x)^(1/3) / x^2 dx

Solution Steps

To solve the integral \(\int \frac{\sqrt[3]{3+1 / x}}{x^{2}} \, dx\), we can use a substitution method. Let's set \(u = 3 + \frac{1}{x}\), then find \(du\) in terms of \(dx\) and substitute back into the integral.

We are given the integral: \[ \int \frac{\sqrt[3]{3 + \frac{1}{x}}}{x^2} \, dx \]

Let \( u = 3 + \frac{1}{x} \). Then, we have: \[ \frac{du}{dx} = -\frac{1}{x^2} \implies du = -\frac{1}{x^2} \, dx \implies dx = -x^2 \, du \]

Substitute \( u \) and \( dx \) into the integral: \[ \int \frac{\sqrt[3]{u}}{x^2} \cdot (-x^2) \, du = -\int u^{1/3} \, du \]

Integrate with respect to \( u \): \[ -\int u^{1/3} \, du = -\frac{3}{4} u^{4/3} + C \]

Substitute \( u = 3 + \frac{1}{x} \) back into the result: \[ -\frac{3}{4} \left(3 + \frac{1}{x}\right)^{4/3} + C \]

Final Answer