Reverse Power Rule (Integration)

The Reverse Power Rule states that for any real number n ≠ -1, the integral of x^n is given by: \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \) where C is the constant of integration.

C = 20 - 27 = -7, so f(x) = 3x^2 - 7.

The constant of integration (C) represents an arbitrary constant added to the indefinite integral, accounting for all possible antiderivatives.

To find the definite integral, evaluate the indefinite integral at the upper limit and subtract the value at the lower limit: \( \int_a^b f(x)dx = F(b) - F(a) \) where F is the antiderivative of f.

The integral of a constant k is given by: \( \int k \, dx = kx + C \) where C is the constant of integration.

The power rule states that if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).