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Antiderivatives & Indefinite Integrals

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Antiderivatives & Indefinite Integrals

Reverse the differentiation process to find families of antiderivative functions.

An antiderivative of f(x) is any function F(x) such that F'(x) = f(x). Since the derivative of a constant is zero, every function has infinitely many antiderivatives differing by a constant C. The indefinite integral notation captures this family.

∫f(x) dx=F(x)+Cwhere F′(x)=f(x)\int f(x)\,dx = F(x) + C \quad \text{where } F'(x) = f(x)
∫xn dx=xn+1n+1+C(n≠−1)∫1x dx=ln⁡∣x∣+C\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1) \qquad \int \frac{1}{x}\,dx = \ln|x| + C
∫ex dx=ex+C∫sin⁡x dx=−cos⁡x+C∫cos⁡x dx=sin⁡x+C\int e^x\,dx = e^x + C \qquad \int \sin x\,dx = -\cos x + C \qquad \int \cos x\,dx = \sin x + C

Basic Antiderivative Rules

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C∫ eˣ dx = eˣ + C∫ sin x dx = −cos x + C∫ cos x dx = sin x + C∫ 1/x dx = ln|x| + C
🔢Finding an Antiderivative
∫(3x² − 4x + 7) dx = 3·(x³/3) − 4·(x²/2) + 7x + C = x³ − 2x² + 7x + C.
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Remember This!

Always verify an antiderivative by differentiating: if you differentiate F(x) + C and get f(x), your antiderivative is correct. This is the ultimate check.

✏️ Try It!

What is ∫(6x² + 4x − 1) dx?

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