Introduction to antiderivatives and integration

Working backwards

Differentiation finds a rate from a function. Integration goes the other way: from a rate back to the function.

An antiderivative of f(x) is a function F(x) with F′(x) = f(x).

Example: An antiderivative of 2x is x², because d/dx x² = 2x.

The constant of integration

x², x² + 5 and x² − 17 all have derivative 2x. So the general antiderivative is:

x² + C

C is the constant of integration.

Notation: the indefinite integral

∫ f(x) dx = F(x) + C

• ∫ is the integral sign
• f(x) is the integrand
• dx shows the variable

This is called an indefinite integral.

Basic rules

Constant multiple and sum rules work as for derivatives.

Examples

• ∫ x⁴ dx = x⁵/5 + C
• ∫ (6x² − 4x + 3) dx = 2x³ − 2x² + 3x + C
• ∫ √x dx = ∫ x^(1/2) dx = (2/3)x^(3/2) + C
• ∫ 1/x³ dx = ∫ x⁻³ dx = −1/(2x²) + C
• ∫ 5eˣ dx = 5eˣ + C

Check: differentiate your answer; you should get the integrand.

Initial conditions

An initial condition pins down C.

Example: f′(x) = 3x² + 2, f(1) = 5. Find f(x).

• f(x) = x³ + 2x + C
• 5 = 1 + 2 + C → C = 2
• f(x) = x³ + 2x + 2

Motion

• Position = ∫ velocity
• Velocity = ∫ acceleration

Example: a(t) = −9.8, v(0) = 15, s(0) = 2

• v(t) = −9.8t + 15
• s(t) = −4.9t² + 15t + 2

Worked examples

Example 1: ∫ 7 dx = 7x + C.

Example 2: ∫ (x³ + x) dx = x⁴/4 + x²/2 + C.

Example 3: ∫ cos x dx = sin x + C.

Common mistakes

• Forgetting + C.
• Power rule errors: add 1 to the exponent, then divide by the new exponent.
• Using the power rule for x⁻¹ (it gives ln|x|).

Try it yourself

1. ∫ 4x³ dx
2. ∫ (2x − 5) dx
3. ∫ x^(−2) dx
4. ∫ 3e^(2x) dx
5. f′(x) = 6x, f(0) = 4. Find f(x).

Answers: 1) x⁴ + C 2) x² − 5x + C 3) −1/x + C 4) (3/2)e^(2x) + C 5) 3x² + 4

Quick recap

• An antiderivative reverses differentiation; integration finds it.
• Always add the constant of integration C in an indefinite integral.
• Use the power rule for integration and initial conditions to find specific functions.