What is ∫ (x^3) dx?

To find the integral of ( x^3 ), we use the power rule for integration, which states that for a function ( x^n ), the integral is given by:

[

\int x^n , dx = \frac{x^{n+1}}{n+1} + C \quad \text{(for ( n \neq -1 ))}

]

In this case, ( n = 3 ). Applying the power rule:

1. Increase the exponent by 1: ( 3 + 1 = 4 ).

2. Divide by the new exponent: ( \frac{x^4}{4} ).

Therefore, the integral of ( x^3 ) is:

[

\int x^3 , dx = \frac{x^4}{4} + C

]

This matches the first option given: ( (1/4)x^4 + C ). However, the choice indicated as the answer was ( (1/3)x^4 + C ), which is not correct since it does not accurately follow the power rule.

The correct answer should indeed be ( (1/4)x^4 + C ), reinforcing the process of applying