Teach me integration

Integration is the reverse process of differentiation, also known as finding the antiderivative. It is used to calculate the accumulation of quantities, such as the area under a curve. An indefinite integral results in a family of functions, always including a constant of integration $C$.

The most common rule for integrating a power of $x$ is:

Let's integrate $f(x) = 4x^3$.

Applying the power rule:

The indefinite integral of $4x^3$ is $\boxed{x^4 + C}$.

A definite integral calculates the exact value of the area under a curve between two specific limits of integration, say from $a$ to $b$. It does not include a constant of integration. The formula is:

where $F(x)$ is the antiderivative of $f(x)$.

Let's find the definite integral of $f(x) = 4x^3$ from $x=1$ to $x=3$.
First, we use the antiderivative $F(x) = x^4$ (from Step 3, without $C$).
Now, evaluate $F(b) - F(a)$:

The definite integral is $\boxed{80}$.