Calculus Chapter 7: Indefinite Integrals

Study Notes

A function F(x) is an antiderivative of f(x) if F'(x) = f(x)
The indefinite integral of f(x) is denoted as ∫f(x)dx and represents the set of all antiderivatives of f(x)
The constant of integration (C) is added to the antiderivative, as the derivative of C is 0
Properties of indefinite integrals:
- Linearity: ∫af(x)dx = a∫f(x)dx and ∫[f(x) + g(x)]dx = ∫f(x)dx + ∫g(x)dx
- Sum and difference: ∫[f(x) ± g(x)]dx = ∫f(x)dx ± ∫g(x)dx

Integration by parts formula: ∫udv = uv - ∫vdu
Used to integrate products of functions, where one function can be easily integrated and the other has a known derivative
Steps to apply integration by parts:
1. Choose u and dv, such that u is a function that can be easily integrated and dv is a function that has a known derivative
2. Differentiate u to get du, and integrate dv to get v
3. Apply the integration by parts formula
4. Simplify the result

Improper integrals are used to extend the concept of definite integrals to functions that have infinite limits or are not defined at certain points
Types of improper integrals:
- Infinite limits: ∫[a,∞)f(x)dx or ∫(-∞,b]f(x)dx
- Discontinuous functions: ∫[a,b]f(x)dx, where f(x) is not defined at a or b
Convergence of an improper integral:
- If the limit exists, the improper integral is said to converge
- If the limit does not exist, the improper integral is said to diverge

Integration by substitution (also known as u-substitution): ∫f(x)dx = ∫f(g(u))g'(u)du
Used to integrate functions that can be transformed into a more easily integrable form
Steps to apply integration by substitution:
1. Choose a substitution u = g(x) that transforms the integral into a more easily integrable form
2. Find the derivative of u, g'(u) = du/dx
3. Replace x with g(u) and dx with g'(u)du in the original integral
4. Simplify the result and evaluate the new integral

An antiderivative F(x) of a function f(x) satisfies F'(x) = f(x)
The indefinite integral ∫f(x)dx represents the set of all antiderivatives of f(x)
The constant of integration C is added to the antiderivative, as C' = 0
The linearity property of indefinite integrals allows for scalar multiplication and addition of integrals

The integration by parts formula is ∫udv = uv - ∫vdu
Integration by parts is used to integrate products of functions, where one function can be easily integrated and the other has a known derivative
The formula requires choosing u and dv, differentiating u, and integrating dv

Improper integrals extend the concept of definite integrals to functions with infinite limits or discontinuities
There are two types of improper integrals: infinite limits and discontinuous functions
An improper integral converges if the limit exists, and diverges if the limit does not exist

The integration by substitution formula is ∫f(x)dx = ∫f(g(u))g'(u)du
Integration by substitution is used to transform functions into a more easily integrable form
The method requires choosing a substitution u = g(x), finding the derivative g'(u), and replacing x with g(u) and dx with g'(u)du