Questions: [ int x^4/3 d x ]

Solution Steps

To evaluate the integral \(\int x^{\frac{4}{3}} \, dx\), we can use the power rule for integration. The power rule states that \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration. In this case, \(n = \frac{4}{3}\).

We start with the integral we want to evaluate: \[ \int x^{\frac{4}{3}} \, dx \]

Using the power rule for integration, we have: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \] In our case, \(n = \frac{4}{3}\). Therefore, we calculate \(n + 1\): \[ n + 1 = \frac{4}{3} + 1 = \frac{4}{3} + \frac{3}{3} = \frac{7}{3} \]

Now we can substitute \(n + 1\) back into the power rule: \[ \int x^{\frac{4}{3}} \, dx = \frac{x^{\frac{7}{3}}}{\frac{7}{3}} + C = \frac{3}{7} x^{\frac{7}{3}} + C \]

Final Answer