Fix the mistake: Integration Methods I (Indefinite Integrals, Integration by Substitution, and Integration by Parts)

Definition and Basic Formulas of Indefinite Integrals

A function F(x) satisfying F'(x) = f(x) is called an antiderivative of f(x), and
　∫ f(x) dx = F(x) + C　（C is the constant of integration）is written. This is the indefinite integral。

Basic formulas：

  • ∫ x^n dx = x^{n+1}/(n+1) + C　（n ≠ −1）
  • ∫ 1/x dx = log|x| + C
  • ∫ e^x dx = e^x + C
  • ∫ sin x dx = −cos x + C、∫ cos x dx = sin x + C
  • ∫ 1/cos²x dx = tan x + C
Example ① Calculate the following.
(1) ∫ (3x²−4x+1) dx　(2) ∫ √x dx　(3) ∫ (2/x + e^x) dx
Solution:
(1) x³ − 2x² + x + C
(2) ∫ x^{1/2} dx = (2/3)x^{3/2} + C = (2/3)x√x + C
(3) 2 log|x| + e^x + C
Key Point
  • Integration is the inverse operation of differentiation
  • Always write +C (the constant of integration)
  • When n = −1 in x^n, use log|x|

Practice Problems

  1. ∫ (4x³ − 6x + 2) dx
  2. ∫ (3sin x − 2cos x) dx

Solutions and Explanations

  1. Solution:x⁴ − 3x² + 2x + C
  2. Solution:−3cos x − 2sin x + C

Math (Math III): related lessons