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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.

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