ºÝºÝߣ

ºÝºÝߣShare a Scribd company logo

Math22 Lecture1

?Download as PPT, PDF?
?
H
hdsierra

This document discusses integration and antiderivatives. It provides the basic rules for finding derivatives and antiderivatives, such as the power rule and how to integrate constants and sums. It then gives examples of finding antiderivatives using these rules, such as integrating x^2 or (4x-1)(2x+3). Students are provided exercises to practice finding antiderivatives of various functions.

1 of 6
Downloaded 22 times
Math 22: Integral Calculus INTEGRATION: The Antiderivative Recall: Basic theorems of finding the derivative f(x) =k ? f¡¯(x) = 0. f(x) = x n ? f¡¯(x) = nx n-1 f(x) = kg(x) ? f¡¯(x) = kg¡¯(x) f(x) = g(x) + h(x) ? f¡¯(x)=g¡¯(x) + h¡¯(x) f(x) = g(x)h(x) ? f¡¯(x) = g(x)h¡¯(x)+g¡¯(x)h(x) f(x) = g(x)/h(x) ? (h(x)g¡¯(x)-g(x)h¡¯(x))/[h(x)] 2
Math 22: Integral Calculus INTEGRATION: The Antiderivative Illustrations: Find the derivative f(x) = 2x +1 h(x) = x 3 +4x 2 -3 g(x) = (x-3)(x 2 +1) f(x) = (3x-1)/(x 3 -2) g(x) = (x 3 +4x 2 -3)(2x +1)
Math 22: Integral Calculus INTEGRATION: The Antiderivative A function F is called an antiderivative of the function f on an interval I if F¡¯(x)=f(x) for all values of x in I. Consequently, we write ? f(x)dx = F(x)+C F(x) + C is the general antiderivative and C is the constant of integration.
Math 22: Integral Calculus INTEGRATION: The Antiderivative Basic theorems of integration: ? dx = x + C ? x n dx = + C ? af(x)dx = a ? f(x) dx ? [f(x) + g(x)]dx = ? f(x)dx + ? g(x)dx
Math 22: Integral Calculus INTEGRATION: The Antiderivative Illustrations: Find the antiderivative ? x 2 dx ? (x -2 +x 3 ) dx ? (4x-1)(2x+3) dx ? ? x(x-5) dx ? dx
Math 22: Integral Calculus INTEGRATION: The Antiderivative Exercises: Find the antiderivative ? x 3 +5x -2 dx ? (3x 2 -5x+4) dx ? (4x+1)(2x-1) dx ? ? x(x 2 -3x) dx ? dx

More Related Content

Math22 Lecture1

  • 1. Math 22: Integral Calculus INTEGRATION: The Antiderivative Recall: Basic theorems of finding the derivative f(x) =k ? f¡¯(x) = 0. f(x) = x n ? f¡¯(x) = nx n-1 f(x) = kg(x) ? f¡¯(x) = kg¡¯(x) f(x) = g(x) + h(x) ? f¡¯(x)=g¡¯(x) + h¡¯(x) f(x) = g(x)h(x) ? f¡¯(x) = g(x)h¡¯(x)+g¡¯(x)h(x) f(x) = g(x)/h(x) ? (h(x)g¡¯(x)-g(x)h¡¯(x))/[h(x)] 2
  • 2. Math 22: Integral Calculus INTEGRATION: The Antiderivative Illustrations: Find the derivative f(x) = 2x +1 h(x) = x 3 +4x 2 -3 g(x) = (x-3)(x 2 +1) f(x) = (3x-1)/(x 3 -2) g(x) = (x 3 +4x 2 -3)(2x +1)
  • 3. Math 22: Integral Calculus INTEGRATION: The Antiderivative A function F is called an antiderivative of the function f on an interval I if F¡¯(x)=f(x) for all values of x in I. Consequently, we write ? f(x)dx = F(x)+C F(x) + C is the general antiderivative and C is the constant of integration.
  • 4. Math 22: Integral Calculus INTEGRATION: The Antiderivative Basic theorems of integration: ? dx = x + C ? x n dx = + C ? af(x)dx = a ? f(x) dx ? [f(x) + g(x)]dx = ? f(x)dx + ? g(x)dx
  • 5. Math 22: Integral Calculus INTEGRATION: The Antiderivative Illustrations: Find the antiderivative ? x 2 dx ? (x -2 +x 3 ) dx ? (4x-1)(2x+3) dx ? ? x(x-5) dx ? dx
  • 6. Math 22: Integral Calculus INTEGRATION: The Antiderivative Exercises: Find the antiderivative ? x 3 +5x -2 dx ? (3x 2 -5x+4) dx ? (4x+1)(2x-1) dx ? ? x(x 2 -3x) dx ? dx