Calculus Chapter 7: Indefinite Integrals

Indefinite Integrals

• 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

• 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

• 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

• 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

Indefinite Integrals

• 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

Integration By Parts

• 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

• 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

Integration By Substitution

• 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