b) \( \int \frac{x^{3}}{6} d x \)

To solve the integral \( \int \frac{x^{3}}{6} dx \), you can take advantage of the power rule for integration. This means you increase the exponent of \( x \) by 1 and divide by the new exponent. In this case, you'll have: \[ \frac{1}{6} \cdot \frac{x^{4}}{4} + C = \frac{x^{4}}{24} + C \] where \( C \) is the constant of integration. So the final answer to your integral is: \[ \frac{x^{4}}{24} + C \] This integral doesn't just yield a polynomial; it showcases the beauty of integral calculus and the idea of reversing differentiation, creating an entirely new function from the original! Now, don’t forget that proper notation matters! When you present your integral results, including the constant of integration \( C \) reminds everyone that there’s an infinite number of functions that can produce the same derivative. A little presentation flair can go a long way in getting your mathematical message across effectively!