Week 2 – Part 1

Last week in Calculus 2, we finished Chapter 5 in Stewart's Calculus. Two of the most important concepts this week were indefinite integrals and u-substitution. Since this will be quite long, I am going to write two posts, each covering one topic.

Indefinite Integrals

An indefinite integral simply represents a "family" of antiderivatives. Suppose $F$ is an antiderivative of $f$. In other words, $F' = f$. Then
$$\int f(x) dx = F(x) + C$$
The $dx$ tells us that we are integrating with respect to $x$, as opposed to another variable. We went over many common indefinite integrals, including this very important one.
$$\int x^n dx = \frac{x^{n+1}}{n+1}$$
for $n\not=-1$. Our professor calls this process "raise and divide." Raise the power $n$ by $1$, and then divide $x^{n+1}$ by the new power $n+1$.

There are many other common indefinite integrals or antiderivatives, which can be found in any calculus textbook. It is very helpful to memorize and know them well. Trust me, that really helps when evaluating integrals.

Stay tuned for my next post!