Antiderivatives and Indefinite Integration
(Study Guide for L-H-E Calculus 7e section 4.1)

Summary:  Finding an antiderivative or indefinite integral is easy if you can think backwards: what function would have this as a derivative? If a function has an antiderivative, then it has infinitely many antiderivatives, all the same except for an added constant.

Key Ideas

• Antiderivative is reverse of derivative: if we say that F is an antiderivative of a known function f, then we mean that F′(x) = f(x)

• Antiderivatives are not unique: 3x², 3x²–11, 3x²+8 are all antiderivatives of 6x because 6x is the derivative of all of them.

• All antiderivatives are the same except for an additive constant, usually indicated by C.
• Andiderivatives solve differential equations. Example:

   = 6x ⇒ dy = 6x dx ⇒ y = ∫ 6x dx = 3x² + C

• The antiderivative is also called the indefinite integral, and ∫ is the integral sign.
• Know the rules on page 244, especially the power rule. All are just rewrites of familiar derivative rules. Example: ∫ 6(x²+7x) dx = 2x³+21x²+C — note that constant is C not 6C.
• You can always verify an indefinite integral or antiderivative by taking the derivative to get the original function.
• Rewrite the integrand to make integrating easier. Example (page 246 example 5):

   = ∫ x^½ dx + ∫ x^-½ dx

• Know how to solve with initial conditions. Example: F′(x) = 4x−7, F(3) = 11.

  F(x) = 2x²−7x+C; substitute 11 = 2(3²)−7(3)+C ⇒ C = 14; answer F(x) = 2x²−7x+14

Terms You Must Know

• antiderivative, an (page 242)
• antiderivative, the general (page 243)
• antidifferentiation (page 243)
• constant of integration (page 243)
• differential equation (page 243)
• indefinite integral and indefinite integration (page 243)
• initial condition (page 247)
• integrand (page 243)
• particular solution (page 247)
• variable of integration (page 243)