Power Rule for Integration: Step-by-Step Guide
The power rule for integration is the inverse of the power rule for differentiation: to integrate x^n (with n not equal to -1), add 1 to the exponent and divide by the new exponent. The special case n = -1 gives ln|x| instead.
Math Racer skill guide · Calculus 2 · AP Calc BC / College
How to solve it, step by step
- Rewrite the integrand as a sum of powers of x. Convert radicals to fractional exponents and reciprocal terms to negative exponents so each term has the form c*x^n.\[\sqrt[m]{x^k} = x^{k/m},\;\; \frac{1}{x^k} = x^{-k}\]
- Apply the power rule term by term. For each term c*x^n with n not equal to -1, the antiderivative is c*x^(n+1)/(n+1). For n = -1, use ln|x|.\[\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \;\;(n \neq -1)\]
- Add constants and combine. Sum the antiderivatives of each term and add a single constant of integration C.\[\int \sum_i c_i x^{n_i}\,dx = \sum_i \frac{c_i x^{n_i+1}}{n_i+1} + C\]
- Verify by differentiating. Differentiate your answer using the power rule and confirm you recover the original integrand.\[\frac{d}{dx}\left[\frac{x^{n+1}}{n+1}\right] = x^n\]
Worked example
Evaluate the integral of (3x^2 + 4*sqrt(x) - 1/x^3) dx.
Step 1: Rewrite as 3x^2 + 4x^(1/2) - x^(-3).
Step 2: Apply power rule term by term. For 3x^2: 3 * x^3/3 = x^3. For 4x^(1/2): 4 * x^(3/2)/(3/2) = (8/3) x^(3/2). For -x^(-3): -x^(-2)/(-2) = x^(-2)/2.
Step 3: Combine: x^3 + (8/3) x^(3/2) + 1/(2x^2) + C.
Step 4: Verify by differentiating: 3x^2 + (8/3)(3/2) x^(1/2) - 1/x^3 = 3x^2 + 4 sqrt(x) - 1/x^3. Matches.
Final answer: x^3 + (8/3) x^(3/2) + 1/(2x^2) + C.
Tips and common mistakes
- Always rewrite radicals as fractional exponents and reciprocals as negative exponents before integrating.
- The power rule fails when n = -1; use ln|x| in that case.
- Add 1 to the exponent first, then divide by the new exponent. Many students do it backwards.
- Constants pass straight through the integral: integral of c f(x) dx = c integral of f(x) dx.
- Sums integrate term by term; you do not need to factor.
- Don't forget the +C; without it, the answer is incomplete on indefinite integrals.
- Verify every answer by differentiating - it takes seconds and catches most errors.