← Back to learning objective

General antiderivative of a power function with negative exponent

Let g(x)=6x4g(x) = \dfrac{6}{x^4}. Find the general antiderivative G(x)G(x) of gg, and write your final answer using a positive exponent (no negative or fractional exponents left in the final simplified form).

Hint

Rewrite 6x4\dfrac{6}{x^4} as 6x−46x^{-4} first, then apply the reverse power rule ∫xn dx=xn+1n+1+C\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C for n≠−1n \neq -1. Don't forget the constant of integration.

Step-by-step solution

First, rewrite the function using a negative exponent: g(x)=6x4=6x−4g(x) = \dfrac{6}{x^4} = 6x^{-4}. This form makes it easier to apply the power rule for antiderivatives, since n=−4≠−1n = -4 \neq -1.

Apply the reverse power rule: ∫6x−4 dx=6⋅x−3−3+C=−2x−3+C\int 6x^{-4}\,dx = 6\cdot \dfrac{x^{-3}}{-3} + C = -2x^{-3} + C. Rewriting with a positive exponent gives G(x)=−2x3+CG(x) = -\dfrac{2}{x^3} + C.

Related problems

Practice similar Calculus problems interactively.

Start Closing the Gap