Indefinite Integration
Indefinite integration allows us to find antiderivatives of functions. It's essentially the reverse process of differentiation.
Basic Indefinite Integrals
Some common indefinite integrals that students should memorize are:
- $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (where $n \neq -1$)
- $\int \sin x dx = -\cos x + C$
- $\int \cos x dx = \sin x + C$
- $\int \frac{1}{x} dx = \ln |x| + C$
- $\int e^x dx = e^x + C$
You can often derive them by using first principles.
NoteThe '+C' at the end of each integral represents the constant of integration. This is crucial because indefinite integrals represent a family of functions that differ only by a constant.
Composites with Linear Functions
These basic integrals can be extended to include composites with linear functions of the form $ax + b$. For example:
- $\int (ax + b)^n dx = \frac{(ax + b)^{n+1}}{a(n+1)} + C$ (where $n \neq -1$)
- $\int \sin(ax + b) dx = -\frac{1}{a}\cos(ax + b) + C$
- $\int \cos(ax + b) dx = \frac{1}{a}\sin(ax + b) + C$
- $\int \frac{1}{ax + b} dx = \frac{1}{a}\ln |ax + b| + C$
- $\int e^{ax + b} dx = \frac{1}{a}e^{ax + b} + C$
Let's integrate $\int (2x + 3)^4 dx$:
Using the formula above with $a=2$, $b=3$, and $n=4$:
$\int (2x + 3)^4 dx = \frac{(2x + 3)^{4+1}}{2(4+1)} + C = \frac{(2x + 3)^5}{10} + C$
Reverse Chain Rule (Integration by Inspection)
The reverse chain rule is a technique used when we can spot a function and its derivative within the integrand. It's based on the chain rule of differentiation but applied in reverse.
The general form is:
$\int k g'(x) f(g(x)) dx = k F(g(x)) + C$
Where $F(x)$ is the antiderivative of $f(x)$ and $C$ is an arbitrary constant.
