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Antiderivative of a Function

Antiderivative of a Function

A function that has a given function as its derivative. For example, F(x) = x3 – 8 is an antiderivative of f(x) = 3x2.

 

 

See also

Indefinite integral

Key Formula

If F′(x)=f(x), then F(x) is an antiderivative of f(x).\text{If } F'(x) = f(x), \text{ then } F(x) \text{ is an antiderivative of } f(x). General antiderivative: F(x)+C\text{General antiderivative: } F(x) + C
Where:
  • F(x)F(x) = An antiderivative — a function whose derivative is f(x)
  • f(x)f(x) = The original function you start with
  • CC = An arbitrary constant, since the derivative of any constant is zero
  • F′(x)F'(x) = The derivative of F(x)

Worked Example

Problem: Find the general antiderivative of f(x) = 4x³.
Step 1: Ask: what function, when differentiated, gives 4x³? Use the power rule for antiderivatives: increase the exponent by 1 and divide by the new exponent.
∫4x3 dx=4⋅x3+13+1=4⋅x44\int 4x^3\,dx = 4 \cdot \frac{x^{3+1}}{3+1} = 4 \cdot \frac{x^4}{4}
Step 2: Simplify the expression.
4⋅x44=x44 \cdot \frac{x^4}{4} = x^4
Step 3: Add the constant of integration C, because any constant disappears when you differentiate.
F(x)=x4+CF(x) = x^4 + C
Step 4: Verify by differentiating F(x). The derivative of x⁴ is 4x³, and the derivative of C is 0.
F′(x)=4x3=f(x)  ✓F'(x) = 4x^3 = f(x) \;\checkmark
Answer: The general antiderivative of f(x) = 4x³ is F(x) = x⁴ + C.

Another Example

Problem: Find the general antiderivative of f(x) = 6x² − 2x + 5.
Step 1: Apply the power rule term by term. For 6x², increase the exponent to 3 and divide by 3.
6⋅x33=2x36 \cdot \frac{x^3}{3} = 2x^3
Step 2: For −2x (which is −2x¹), increase the exponent to 2 and divide by 2.
−2⋅x22=−x2-2 \cdot \frac{x^2}{2} = -x^2
Step 3: For the constant 5 (which is 5x⁰), increase the exponent to 1 and divide by 1.
5⋅x11=5x5 \cdot \frac{x^1}{1} = 5x
Step 4: Combine all terms and add the constant of integration.
F(x)=2x3−x2+5x+CF(x) = 2x^3 - x^2 + 5x + C
Answer: The general antiderivative is F(x) = 2x³ − x² + 5x + C.

Frequently Asked Questions

What is the difference between an antiderivative and an indefinite integral?
They refer to essentially the same thing. An antiderivative is any single function F(x) whose derivative is f(x). The indefinite integral, written ∫f(x) dx, represents the entire family of antiderivatives, which is F(x) + C. The notation with the integral sign and the + C emphasizes that there are infinitely many antiderivatives differing by a constant.
Why do you add + C when finding an antiderivative?
The derivative of any constant is zero, so if F(x) is an antiderivative of f(x), then F(x) + 7, F(x) − 3, or F(x) + C for any constant C is also an antiderivative. The + C accounts for all these possibilities. Without it, you are giving only one specific antiderivative rather than the complete general solution.

Antiderivative vs. Derivative

These are inverse operations. The derivative of F(x) gives you f(x) — it tells you the rate of change. The antiderivative starts with f(x) and recovers F(x) — it reverses that process. Differentiation is unique (there is exactly one derivative), but antidifferentiation produces a family of functions differing by a constant C.

Why It Matters

Antiderivatives are the foundation of integral calculus. They allow you to compute areas under curves, total distances from velocity functions, and accumulated quantities in physics and engineering. The Fundamental Theorem of Calculus directly connects antiderivatives to definite integrals, making them essential for evaluating integrals without resorting to limit-of-sums calculations.

Common Mistakes

Mistake: Forgetting the constant of integration + C.
Correction: Every antiderivative problem has infinitely many solutions differing by a constant. Always write + C in your general antiderivative. Omitting it means your answer is incomplete — it's just one particular antiderivative, not the general one.
Mistake: Applying the power rule incorrectly by forgetting to increase the exponent before dividing.
Correction: The antiderivative of xⁿ is x^(n+1)/(n+1), not xⁿ/n. First add 1 to the exponent, then divide by that new exponent. For example, the antiderivative of x² is x³/3, not x²/2.

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