What is the formula for integrating (x^n) dx?

The integration of a function of the form ( x^n ) is a fundamental concept in calculus, particularly in the context of definite and indefinite integrals. The correct formula for integrating ( x^n ) with respect to ( x ) is derived using the power rule for integration.

According to the power rule, if ( n ) is not equal to -1, the integral of ( x^n ) dx is calculated by increasing the exponent by 1 and dividing by the new exponent. This gives us:

[

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

]

Where ( C ) is the constant of integration. This formula applies directly when the exponent ( n ) is anything except -1. If ( n = -1 ), integrating ( x^{-1} ) leads to the natural logarithm function.

This understanding helps in solving a wide variety of problems in calculus, including finding the area under curves and solving differential equations. Familiarity with this rule is essential for further studies in mathematics and physics as well.

The other choices do not follow the established power rule for integration. The approach in those responses either misap