What is the integral of x^n where n ≠ -1?

The integral of ( x^n ) where ( n \neq -1 ) is derived from the power rule for integrals. According to this rule, when you integrate a function of the form ( x^n ), you increase the power by one and then divide by the new power.

Thus, integrating ( x^n ) results in:

[

\int x^n , dx = \frac{1}{n+1} x^{n+1} + C

]

Here, ( n+1 ) is the new exponent after increasing ( n ) by one. Dividing by ( n+1 ) ensures that the area under the curve represented by ( x^n ) is accurately represented in the antiderivative. The ( C ) represents the constant of integration, which accounts for any vertical shifts in the family of antiderivative functions.

The other choices do not follow this rule. For instance, the option suggesting ( \frac{1}{n} x^n + C ) does not apply since it does not reflect the appropriate adjustment in the exponent and division needed for finding the antiderivative of ( x^n ). Similarly, the choices involving