Prove the formula for the integral of x^n.

The formula for the integral of x^n is (x^(n+1))/(n+1) + C, where C is the constant of integration.

To prove this formula, we can use the power rule of integration. According to this rule, the integral of x^n is (x^(n+1))/(n+1) + C, where C is the constant of integration. We can verify this formula by differentiating it and checking if we get back the original function.

Let's differentiate (x^(n+1))/(n+1) + C with respect to x. Using the power rule of differentiation, we get:

d/dx [(x^(n+1))/(n+1) + C] = (n+1)x^n

This is the same as the original function x^n, except for the constant factor (n+1). Therefore, we can conclude that the formula (x^(n+1))/(n+1) + C is indeed the antiderivative of x^n.

Another way to prove this formula is to use integration by substitution. Let u = x^(n+1), then du/dx = (n+1)x^n. Rearranging, we get dx = du/((n+1)x^n). Substituting these into the integral of x^n, we get:

∫x^n dx = ∫u du/((n+1)x^n)

= (1/(n+1)) ∫u^(-n/(n+1)) du

= (1/(n+1)) u^((n+1)/(n+1)) + C

= (x^(n+1))/(n+1) + C

This is the same as the formula we derived earlier using the power rule of integration. Therefore, we have proven the formula for the integral of x^n.

For further understanding, you can explore various integration techniques, which can be applied to different types of functions. Additionally, the concepts of definite and indefinite integrals are foundational to understanding the broader context of integration in calculus. For a solid basis on how these rules are applied, reviewing the basic integration rules can also be beneficial.