Power rule for integrals

Here is another elementary use of the fundamental theorem of calculus.

Lemma: Power rule for integrals

if n ≠ -1, then ∫ axⁿdx = (1/[n+1])axⁿ⁺¹+C

Proof:

(1) d/dx[ (1/[n+1])axⁿ⁺¹+C] = (1/[n+1]a)d/dx[xⁿ⁺¹] + d/dx[C] =
(1/[n+1])(n+1)*axⁿ + 0 = axⁿ

First, we apply Lemma 3, here to split up the sum. Since the derivative is defined in terms of limits (see Definition 1, here), we can move the constants (1/[n+1])*a. The constant rule (see Lemma 1, here) gives us that d/dx[C] = 0. Finally, the power rule for derivatives (see Lemma 2, here) gives us our result.

(2) Since (1/[n+1])axⁿ⁺¹ + C is the antiderivative for axⁿ (see Definition 2, here, for definition of antiderivative), the Fundamental Theorem of Calculus (see Theorem 2, here) gives us our conclusion.

QED