An antiderivative reverses differentiation: it finds the function whose derivative is the one you started with. Indefinite integrals are the same idea written with integral notation. You can check your work with the free tools on Calc Solver.
What an Antiderivative Is
If the derivative of x^2 is 2x, then x^2 is an antiderivative of 2x. Every function has many antiderivatives, because adding a constant never changes a derivative. That is why every indefinite integral answer ends with + C, the constant of integration.
Step 1: Reverse the Power Rule
For a power of x, add 1 to the exponent and divide by the new exponent. The antiderivative of x^3 is x^4/4, because differentiating x^4/4 gives back x^3. This reverse power rule handles most basic polynomials term by term.
Step 2: Handle Each Term Separately
Integrate a sum term by term, just as you differentiate term by term. The antiderivative of 3x^2 + 2x is x^3 + x^2 + C. Constants factor out: the antiderivative of 5x^2 is 5 times x^3/3, which is (5/3)x^3, plus C.
Step 3: Check by Differentiating
Always verify by differentiating your answer. If differentiating x^3 + x^2 + C gives 3x^2 + 2x, the antiderivative is correct. This check takes seconds and catches sign errors and exponent mistakes immediately.
Common Functions to Know
The antiderivative of sin(x) is -cos(x) + C, and the antiderivative of cos(x) is sin(x) + C. The antiderivative of e^x is e^x + C, and the antiderivative of 1/x is ln|x| + C. Memorizing these few forms covers a large share of basic problems.
Common Mistakes to Avoid
First, forgetting + C, which makes the answer incomplete. Second, dividing by the wrong exponent when reversing the power rule. Third, trying to integrate a product with the product rule, which does not reverse: products usually need substitution, a separate technique.
Practice Problems to Try
Find the antiderivatives of x^4, 2x + 3, and 3x^2 – 1. Differentiate each answer to confirm it matches the original function.