Antiderivatives and Indefinite Integrals

What You’ll Learn

In this lesson you’ll learn how to go backwards from a derivative to the original function. This is the beginning of integral calculus.

The Concept

An antiderivative of f(x) is a function F(x) whose derivative is f(x). In other words, F’(x) = f(x).

The indefinite integral is the notation for finding all antiderivatives:

$$\int f(x) \, dx = F(x) + C$$

The + C is the constant of integration. It’s there because if F(x) is an antiderivative, so is F(x) + 7, or F(x) - 3, or F(x) + any constant. The derivative of a constant is zero, so you can’t tell which constant was there just from the derivative.

Basic Integration Rules

Power rule (reverse):

$$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$$

Special case for 1/x:

$$\int \frac{1}{x} \, dx = \ln|x| + C$$

Exponential:

$$\int e^x \, dx = e^x + C$$

Trig:

$$\int \sin x \, dx = -\cos x + C$$ $$\int \cos x \, dx = \sin x + C$$

Constants and sums pass through the integral, just like with derivatives.

F(x) = x² + C (C = -2, 0, 2, 4) f(x) = 2x (the derivative) all four curves have the same derivative

The green line is f(x) = 2x, the derivative. The four blue parabolas are all antiderivatives: x² - 2, x², x² + 2, and x² + 4. They’re the same shape, just shifted up or down. All four have the same derivative (2x), which is why you need + C to capture the whole family.

Worked Example

Example 1: Polynomial

Find the integral of 3x² - 4x + 7.

$$\int (3x^2 - 4x + 7) \, dx = x^3 - 2x^2 + 7x + C$$

Each term uses the power rule: add 1 to the exponent, divide by the new exponent. The constant 7 integrates to 7x.

Example 2: Exponential and trig

Find the integral of e to the x plus 2 sin(x).

$$\int (e^x + 2\sin x) \, dx = e^x - 2\cos x + C$$

The integral of e to the x is itself. The integral of sin(x) is negative cos(x), and the 2 passes through.

Example 3: Fractional exponents

Find the integral of x to the 1/2 plus 5x to the negative 3.

$$\int (x^{1/2} + 5x^{-3}) \, dx = \frac{x^{3/2}}{3/2} + 5 \cdot \frac{x^{-2}}{-2} + C = \frac{2}{3}x^{3/2} - \frac{5}{2}x^{-2} + C$$

The power rule works for any exponent except -1. Add 1 to the exponent, divide by the new exponent.

Real-World Application

Antiderivatives connect rates of change back to total quantities:

• Physics: if you know acceleration a(t), integrating gives velocity v(t). Integrating again gives position s(t).
• Economics: if you know marginal cost (the derivative of total cost), integrating gives total cost.
• Biology: if you know the rate of population growth, integrating gives the total population.
• Game dev: if a character’s speed function is known, integrating gives the total distance traveled.

You’ve Got This

Integration is just differentiation in reverse. The power rule flips: instead of “bring down the exponent and subtract 1,” you “add 1 to the exponent and divide by the new exponent.” Always add + C. If you can differentiate, you can integrate. The two skills are mirror images of each other.

Quiz

Every indefinite integral must include:

• A.A limit
• B.A derivative
• C.+ C (constant of integration)
• D.A definite bound

The antiderivative of x⁴ is:

• A.4x³ + C
• B.(1/5)x⁵ + C
• C.x⁵ + C
• D.5x⁴ + C

The integral of e to the x is:

• A.ln(x) + C
• B.x times e to the x + C
• C.e to the x + C
• D.e to the (x+1) + C

The integral of sin(x) is:

• A.cos(x) + C
• B.sin(x) + C
• C.-sin(x) + C
• D.-cos(x) + C

Why is + C necessary in indefinite integrals?

• A.It makes the math harder
• B.Constants have derivative zero, so infinitely many antiderivatives exist
• C.It's optional
• D.It only applies to trig functions