MA 16500: Analytic Geometry and Calculus I

4.9. Antiderivatives

Goal: Understand antiderivatives and indefinite integrals, learn how to compute them using basic rules and standard formulas.

1. Antiderivatives

Definition (Antiderivative):

A function \( F(x) \) is called an antiderivative of \( f(x) \) if

\[ F'(x) = f(x) \]

Example 1.1

Show that \( F(x) = x^2 \) is an antiderivative of \( f(x) = 2x \).

\[ F'(x) = \frac{d}{dx}(x^2) = 2x = f(x) \]

So \( F(x) \) is an antiderivative of \( f(x) \).

Example 1.2

Show that \( F(x) = -\cos x \) is an antiderivative of \( f(x) = \sin x \).

\[ F'(x) = \frac{d}{dx}(-\cos x) = \sin x = f(x) \]

So \( F(x) \) is an antiderivative of \( f(x) \).

Properties of Antiderivatives:

An antiderivative of \( f(x) \) is determined only up to a constant:

  • If \( F(x) \) is an antiderivative of \( f(x) \), then \( F(x) + C \) is also an antiderivative for any constant \( C \).
  • If \( F(x) \) and \( G(x) \) are both antiderivatives of \( f(x) \), then they differ by a constant: \[ G(x) = F(x) + C \]

Indefinite Integral Notation:

\[ \int f(x)\,dx = F(x) + C \quad \text{where } F'(x) = f(x) \]
  • \( \int \) is the integral sign.
  • \( f(x) \) is called the integrand.
  • \( dx \) indicates the variable of integration.
  • \( F(x) \) is an antiderivative of \( f(x) \).
  • \( C \) is the constant of integration.

This expression represents the indefinite integral, which is the family of all antiderivatives of \( f(x) \).

It is read as: “the integral of \( f(x) \) with respect to \( x \)”.

2. Common Antiderivatives

Constant Multiple Rule:

\[ \int c \, f(x) \, dx = c \int f(x) \, dx \quad \text{for any real constant } c \]

Sum Rule:

\[ \int \big(f(x) + g(x)\big) \, dx = \int f(x) \, dx + \int g(x) \, dx \]

Basic Antiderivative Formulas:

\[ \int e^x \, dx = e^x + C \] \[ \int a^x \, dx = \frac{a^x}{\ln a} + C \quad \text{for } a > 0, \, a \ne 1 \] \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \ne -1) \]

Indefinite Integrals of Trigonometric Functions:

\[ \int \sin x \, dx = -\cos x + C \] \[ \int \cos x \, dx = \sin x + C \] \[ \int \sec^2 x \, dx = \tan x + C \] \[ \int \csc^2 x \, dx = -\cot x + C \] \[ \int \sec x \tan x \, dx = \sec x + C \] \[ \int \csc x \cot x \, dx = -\csc x + C \]

Indefinite Integrals of Inverse Trigonometric Functions:

\[ \int \frac{1}{\sqrt{1-x^2}} \, dx = \arcsin x + C \] \[ \int \frac{-1}{\sqrt{1-x^2}} \, dx = \arccos x + C \] \[ \int \frac{1}{1+x^2} \, dx = \arctan x + C \] \[ \int \frac{-1}{1+x^2} \, dx = \operatorname{arccot} x + C \] \[ \int \frac{1}{|x|\sqrt{x^2-1}} \, dx = \operatorname{arcsec} |x| + C \] \[ \int \frac{-1}{|x|\sqrt{x^2-1}} \, dx = \operatorname{arccsc} |x| + C \]

Example:

\[ I = \int \Big( 2\sqrt{x} + 3x^3 - 5\cos x + e^x + \frac{6}{1+x^2} \Big) dx \]
\begin{align*} I &= \int 2\sqrt{x} \, dx + \int 3x^3 \, dx - \int 5\cos x \, dx + \int e^x \, dx + \int \frac{6}{1+x^2} \, dx \\ &= 2 \int x^{1/2} \, dx + 3 \int x^3 \, dx - 5 \int \cos x \, dx + \int e^x \, dx + 6 \int \frac{1}{1+x^2} \, dx \\ &= 2 \cdot \frac{2}{3} x^{3/2} + 3 \cdot \frac{x^4}{4} - 5 (\sin x) + e^x + 6 \arctan x + C\\ &= \frac{4}{3} x^{3/2} + \frac{3}{4} x^4 - 5 \sin x + e^x + 6 \arctan x + C \end{align*}